CN107271958A - The approximate convex optimized algorithm of Multi-target position outer approximation based on arrival time - Google Patents
The approximate convex optimized algorithm of Multi-target position outer approximation based on arrival time Download PDFInfo
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- CN107271958A CN107271958A CN201710720839.6A CN201710720839A CN107271958A CN 107271958 A CN107271958 A CN 107271958A CN 201710720839 A CN201710720839 A CN 201710720839A CN 107271958 A CN107271958 A CN 107271958A
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
- G01S5/00—Position-fixing by co-ordinating two or more direction or position line determinations; Position-fixing by co-ordinating two or more distance determinations
- G01S5/02—Position-fixing by co-ordinating two or more direction or position line determinations; Position-fixing by co-ordinating two or more distance determinations using radio waves
- G01S5/06—Position of source determined by co-ordinating a plurality of position lines defined by path-difference measurements
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
- G01S5/00—Position-fixing by co-ordinating two or more direction or position line determinations; Position-fixing by co-ordinating two or more distance determinations
- G01S5/0009—Transmission of position information to remote stations
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- G06—COMPUTING; CALCULATING OR COUNTING
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Abstract
The invention discloses a kind of approximate convex optimized algorithm of Multi-target position outer approximation based on arrival time, primal problem model is constructed first, then the use of relaxation method is the convex optimization problem model of a MIXED INTEGER by the relaxation of former problem, and new suitable constraint is built for positive semidefinite matrix, then outer approximation approximate data submodel and convex row optimization problem model are used, the coordinate value of each target to be positioned is tried to achieve using OAA algorithms.The present invention is using outside close approximation algorithm, and layout and target region to base station do not have existing method so complicated requirement, and is able to ensure that and converges to global optimum, and does not need initial estimation point.
Description
Technical field
The invention belongs to based on wireless signal field of locating technology, more particularly to a kind of convex optimization of Multi-target position problem
Algorithm, it is adaptable to the Multi-target position problem based on TOA.
Background technology
Wireless location technology, is used for orientation problem in World War II to military targets such as naval vessels and fighter planes earliest.With
The development of science and technology, wireless location technology is more and more widely used in fields such as industrial, civilian and national defence.Such as
Emergency relief is responded, and dangerous goods are followed the trail of, mobile phone positioning, process control.Wireless location algorithm is improved by mathematical method
Estimating speed and precision are always relevant technical personnel's concern.
Have benefited from the development of convex optimized algorithm, efficient convex optimization relaxing techniques can apply to target orientation problem.With
The fast development of the technologies such as big data, mobile Internet and Internet of Things, Multi-target position problem has turned into a wireless location neck
The study hotspot in domain.Current main research is focused on going to improve single goal orientation problem using newest convex optimized algorithm, main
Positive semidefinite relaxation is used, the method such as quadric cone relaxation and branch and bound solves single goal orientation problem.Wherein in order to
Computational efficiency is improved, single order accelerating algorithm, the algorithm of the high convergence rate such as quasi-Newton method and interior point method is also employed in more than calculating
Relaxation after the problem of in.
Especially, for Multi-target position problem, because which target base station is unable to identification source in, therefore this is asked
Topic is a NP-hard problem, and existing Multi-target position algorithm can only be in reasonable Arrangement base station, and then target distribution is in specific model
On the premise of enclosing, select just obtain a globally optimal solution under conditions of suitable estimated initial point.Otherwise, it can only ensure to receive
Hold back to a locally optimal solution.And in order to solve the problems, such as NP-hard, computation complexity is presented with the increase of destination number
Go out exponential increase, and easily sink into locally optimal solution.Therefore designing one kind can improve to base station and target location robust
Property, high convergence rate and converge to the optimal algorithm of universe there is very big actual application prospect.
The content of the invention
It is a kind of many based on arrival time (TOA) it is an object of the invention to propose for above-mentioned the deficiencies in the prior art
Target localized external approaches approximate convex optimized algorithm (OAA), and this method is most prominent based on outside close approximation convex optimized algorithm
Advantage is that, using outside close approximation algorithm, layout and target region to base station do not have existing method so complicated
Requirement, and be able to ensure that and converge to global optimum, and do not need initial estimation point.
What the present invention was realized in：
The technical thought for realizing the present invention is to construct primal problem model (1) first, is then asked original using relaxation method
Topic relaxation is the convex optimization problem model (2) of a MIXED INTEGER, and builds new suitable constraint for positive semidefinite matrix, then
Using outer approximation approximate data submodel (3) and convex row optimization problem model (4), tried to achieve using OAA algorithms to be positioned
The coordinate value of each target.
More specifically step of the invention is as follows：
Multi-target position (time of arrival based multiple source based on arrival time
Localization TOA-MSL) the approximate convex optimized algorithm of outer approximation (outer approximal approach
Algorithm OAA), including relaxed by positive semidefinite, after quadric cone relaxation, Taylor's relaxation and addition SDP constraintss, root
A convex optimization problem of MIXED INTEGER is obtained according to maximal possibility estimation (MLE) model, essence is obtained according to outer approximation approximate data
True multiple target coordinate.
Specifically include following steps：
A) defined by TOA, the primal problem model of a MIXED INTEGER, non-convex is constructed using maximal possibility estimation
(1)；
B) relaxed by using positive semidefinite, Taylor relaxation and quadric cone relaxation etc. relaxation method and addition SDP constraint bar
Part, is the convex optimization problem model (2) of a MIXED INTEGER by the relaxation of former problem；
C) the solution P of an integer part, corresponding optimal function are obtained by solving outer approximation approximate data submodel (3)
It is worth the lower bound (LB) as whole algorithm；
D) fixed integer solution part, b) the convex optimization problem model (2) of corresponding MIXED INTEGER be changed into the optimization of convex row
Problem model (4), continuous part solution, and corresponding functional value, correspondence letter are obtained by solving the convex row optimization problem model (4)
Minimum value in numerical value is the upper bound (UB) of whole algorithm；
E) judge solution that convex row optimization problem model (4) obtains whether be former problem optimal solution：
ⅰ.If UB>LB+constant, d) in continuous solution and the integer solution corresponding with him be not former problem most
Excellent solution, the constraints of the outer approximation approximate data submodel (3) in iteration upgrading b), it is ensured that the integer solution occurred
Will not again occur, continue executing with b)-d) until obtaining optimal solution or iterations more than the default upper limit.
ⅱ.If UB<=LB+constant, d) in continuous solution and the integer solution corresponding with him be former problem most
Excellent solution, shuts down procedure.
Further scheme is：
The step a) includes：
I .TOA is defined, it is assumed that have its one L × K matrix of correspondence of K target x to be positioned, one L × M of M base station s correspondence
The location matrix s of matrix, wherein base station is, it is known that arrival time corresponding K × Metzler matrix d is, it is known that the location matrix x of target is unknown.
Understand that the arrival time of k-th of target to i-th of base station is as follows according to TOA definition：
Wherein, last is noise, without loss of generality, it will be assumed that each noise Gaussian distributed, and
Independent same distribution.In Multi-target position, it is assumed that can not recognize which mesh the signal that he obtains comes from for each base station
Mark (is expressed as), in order to recognize each target, we are that each base station introduces a permutation matrix
(permutation matrix), to recognize each target, method is following, and (for i-th of base station, identification is from k-th of target
TOA values)：
II goes to minimize the absolute value of noise, based on this, we construct following multiple target and determined by maximal possibility estimation
Bit model (1)：
ZK, i>=0, Pi∈ π,
Primal problem model (1)
Wherein, Pi represents the corresponding permutation matrix in i-th of base station；The TOA data received for i-th of base station；Zk, i table
Show square of the actual TOA data of k-th target and i-th base station and the difference of estimated data；For to be calculated k-th
Target and the true TOA values of i-th of base station；Second constraint is TOA physical definition.It will be apparent that the problem is one
The non-convex problem of MIXED INTEGER, existing algorithm is it is difficult to ensure that converge to globally optimal solution.
Further scheme is：
The step b) includes：
I calculates the position of multiple targets according to the TOA data and the positional information of base station；
II carries out including but is not limited to positive semidefinite relaxation, quadric cone relaxation, Taylor's relaxation and addition SDP to primal problem
Constraints with cause former problem be changed into convex problem and reduction computation complexity.
Further scheme is：
The step c) includes：
The positive semi-definite constraints of SDP constraints is fallen in I relaxations, obtains outer approximation approximate data submodel (3)；
II outer approximation approximate data submodels (3) are a convex optimization problems of MIXED INTEGER-quadric cone；
Outer approximation approximate data submodel (3) after III relaxations, can use including but not limited to branch-and-bound calculation
Method, interior point method, fixed step size gradient descent method, variable step-length gradient drop method and Newton method obtain the optimal solution and most of subproblem
Major function value；
IV preserves the integer part of optimal solution, and optimal function value is defined as the lower bound LB of former problem, into step c).
Further scheme is：
The step d) includes：
Behind I fixed integers part, the former convex optimization problem model (2) of MIXED INTEGER becomes in order to which one continuously carries partly just
Conclude a contract or treaty the convex optimization problem of continuously differentiable point of beam and quadric cone constraint；
The II problems can use including but not limited to interior point method, gradient descent method (fixed step size, change step) and ox
The ripe convex optimized algorithm such as method obtains the optimal solution and optimal function value of the subproblem；
III optimal function values are designated as the upper bound (UB) of former problem.
Further scheme is：
The step e) includes：
Ensure that the unduplicated method of integer solution includes but is not limited to：
The dual problem of convex row optimization problem model (4), obtains location variable x X pairs of slack variable during I is calculated d)
The dual optimal solution X_dual and the corresponding slack variable D of true arrival time d dual optimal solution D_dual answered, in son b)
Added about in problem<X,X_dual>>=0 (Kronecker product)；<D,D_dual>>=0.
II addition constraints directly in outer approximation approximate data submodel (3) b)<P,P><=M × K-1. (P is b)
The integer solution obtained in step)
Further scheme is：
Maximal possibility estimation (MLE) model is built, including：
The probability density function that I sets up noise (includes but is not limited to Gaussian distributed)；
The II calculating actual times of arrival are expressed as d0+ε；
III object functions minimize for the quadratic sum of noise；
Noise signal to noise ratio (SNR) substantial scope that IV considers is 0-100dB.
Further scheme is：
Relaxation step in step b), positive semidefinite relaxation and quadric cone relaxation, obtain the maximal possibility estimation after following relaxation
The convex optimization problem model (2) of problem, i.e. MIXED INTEGER
Wherein first constraint is quadric cone relaxation, and second, the 3rd, the 4th and the 5th constraints is partly just
Fixed relaxation, is relaxed by positive semidefinite and the convex optimization problem of a MIXED INTEGER-positive semidefinite-quadric cone was obtained after being relaxed with quadric cone
Model (2).
Further scheme is：
Taylor's relaxation is comprised the following steps that：
By to carry out Taylor's single order relaxation of second constraints in primal problem model (1), second order relaxation or
High-order relaxes, so that a loose single order, second order or high-order inequality are obtained, so that it is whole that problem is converted into a mixing
The convex optimization problem of number-quadric cone.
Further scheme is：
Positive semidefinite matrix (SDP) constraints after the new relaxation of addition, be specially：
Positive semidefinite matrix D after being relaxed to positive semidefinite adds constraints, to ensure diagonal element element in positive semidefinite matrix
Di k,kNumerical value is as close possible to (Di k,L+1)2, so that it is guaranteed that positive semidefinite relaxation is with former problem equivalent, the method used includes but not limited
In the following constraints of addition：
Further scheme is：
Outer approximation approximate data submodel (3), its model is including but not limited to feature as follows：
<Di, Ddual { iternum }i>≥0
<Xk, Xdual { iternum }k>≥0
Outer approximation approximate data submodel (3i)
ZK, i>=0, Pi∈ Π,
Outer approximation approximate data submodel (3ii)
Further scheme is：
Convex row optimization problem model (4), its model is including but not limited to feature as follows：
Convex row optimization problem model (4)
The model is a continuous-convex optimization problem of positive semidefinite-quadric cone, and convex optimized algorithm can be used to obtain optimal
Solution, wherein first constraints represents that the integer solution that will be obtained brings our TOA data into, so that the 0-1 integer variables that disappear.
Further scheme is：
The corresponding positive semidefinite slack variable D of the true arrival time d of D_dual dual optimal solution D_dual, and target location
The corresponding positive semidefinite slack variable X of known variables x dual optimal solution X_dual, can be convex excellent with following characteristics by calculating
Change problem is obtained：
The corresponding dual problem model of relaxation problem
Wherein second and the 3rd constraint distribution represent D_dual and X_dual.
The approximate convex optimized algorithm of Multi-target position outer approximation (OAA) disclosed by the invention based on arrival time (TOA),
With regard to that can obtain a globally optimal solution after the iteration of limited number of time, simulation result shows in extensive Multi-target position problem,
This method remains to obtain globally optimal solution in a short period of time, and when signal to noise ratio is higher than 30dB, this method can reach Cram é r-
Rao lower bounds (CRLB).This method can be applied to mobile phone positioning, and rescue is positioned, in the field such as radar UFO positioning.
Brief description of the drawings
Fig. 1 is schematic flow sheet of the present invention；
Fig. 2 is the target location scatter diagram that the embodiment of the present invention 1 is calculated；
Fig. 3 is the target location scatter diagram that the embodiment of the present invention 2 is calculated.
Embodiment
With reference to embodiment, the present invention is further illustrated.
As shown in Figure 1, a kind of approximate convex optimized algorithm of Multi-target position outer approximation for being based on arrival time (TOA)
(OAA), including by positive semidefinite relax, after quadric cone relaxation, Taylor's relaxation and addition SDP constraintss, according to maximum likelihood
Estimation (MLE) model obtains a convex optimization problem of MIXED INTEGER, and accurate multiple target is obtained according to outer approximation approximate data
Coordinate.
Specifically include following steps：
A) defined by TOA, the primal problem model of a MIXED INTEGER, non-convex is constructed using maximal possibility estimation
(1)；
B) relaxed by using positive semidefinite, Taylor relaxation and quadric cone relaxation etc. relaxation method and addition SDP constraint bar
Part, is the convex optimization problem model (2) of a MIXED INTEGER by the relaxation of former problem；
C) the solution P of an integer part, corresponding optimal function are obtained by solving outer approximation approximate data submodel (3)
It is worth the lower bound (LB) as whole algorithm；
D) fixed integer solution part, b) the convex optimization problem model (2) of corresponding MIXED INTEGER be changed into the optimization of convex row
Problem model (4), continuous part solution, and corresponding functional value, correspondence letter are obtained by solving the convex row optimization problem model (4)
Minimum value in numerical value is the upper bound (UB) of whole algorithm；
E) judge solution that convex row optimization problem model (4) obtains whether be former problem optimal solution：
ⅰ.If UB>LB+constant, d) in continuous solution and the integer solution corresponding with him be not former problem most
Excellent solution, the constraints of the outer approximation approximate data submodel (3) in iteration upgrading b), it is ensured that the integer solution occurred
Will not again occur, continue executing with b)-d) until obtaining optimal solution or iterations more than the default upper limit.
ⅱ.If UB<=LB+constant, d) in continuous solution and the integer solution corresponding with him be former problem most
Excellent solution, shuts down procedure.
Further scheme is：
The step a) includes：
I .TOA is defined, it is assumed that have its one L × K matrix of correspondence of K target x to be positioned, one L × M of M base station s correspondence
The location matrix s of matrix, wherein base station is, it is known that arrival time corresponding K × Metzler matrix d is, it is known that the location matrix x of target is unknown.
Understand that the arrival time of k-th of target to i-th of base station is as follows according to TOA definition：
Wherein, last is noise, without loss of generality, it will be assumed that each noise Gaussian distributed, and solely
Vertical same distribution.In Multi-target position, it is assumed that can not recognize which target the signal that he obtains comes from for each base station
(it is expressed as), in order to recognize each target, we are that each base station introduces a permutation matrix (permutation
Matrix) to recognize each target, method is following (for i-th of base station, recognizing the TOA values from k-th of target)：
II goes to minimize the absolute value of noise, based on this, we construct following multiple target and determined by maximal possibility estimation
Bit model (1)：
ZK, i>=0, Pi∈ π,
Primal problem model (1)
Wherein, PiRepresent the corresponding permutation matrix in i-th of base station；The TOA data received for i-th of base station；Zk,iRepresent
Square of the actual TOA data of k-th target and i-th base station and the difference of estimated data；For k-th of mesh to be calculated
Mark and the true TOA values of i-th of base station；Second constraint is TOA physical definition.Mixed it will be apparent that the problem is one
The non-convex problem of integer is closed, existing location algorithm hardly results in globally optimal solution.
Further scheme is：
The step b) includes：
I calculates the position of multiple targets according to the TOA data and the positional information of base station；
II carries out including but is not limited to positive semidefinite relaxation, quadric cone relaxation, Taylor's relaxation and addition SDP to primal problem
Constraints with cause former problem be changed into convex problem and reduction computation complexity.
Further scheme is：
The step c) includes：
The positive semi-definite constraints of SDP constraints is fallen in I relaxations, obtains outer approximation approximate data submodel (3)；
II outer approximation approximate data submodels (3) are a convex optimization problems of MIXED INTEGER-quadric cone；
Outer approximation approximate data submodel (3) after III relaxations, can use including but not limited to branch-and-bound calculation
Method, interior point method, fixed step size gradient descent method, variable step-length gradient drop method and Newton method obtain the optimal solution and most of subproblem
Major function value；
IV preserves the integer part of optimal solution, and optimal function value is defined as the lower bound LB of former problem, into step c).
Further scheme is：
The step d) includes：
Behind I fixed integers part, the former convex optimization problem model (2) of MIXED INTEGER becomes in order to which one continuously carries partly just
Conclude a contract or treaty the convex optimization problem of continuously differentiable point of beam and quadric cone constraint；
The II problems can use including but not limited to interior point method, gradient descent method (fixed step size, change step) and ox
The ripe convex optimized algorithm such as method obtains the optimal solution and optimal function value of the subproblem；
III optimal function values are designated as the upper bound (UB) of former problem.
Further scheme is：
The step e) includes：
Ensure that the unduplicated method of integer solution includes but is not limited to：
The dual problem of convex row optimization problem model (4), obtains location variable x X pairs of slack variable during I is calculated d)
The dual optimal solution X_dual and the corresponding slack variable D of true arrival time d dual optimal solution D_dual answered, in son b)
Added about in problem<X,X_dual>>=0 (Kronecker product)；<D,D_dual>>=0.
II addition constraints directly in outer approximation approximate data submodel (3) b)<P,P><=M × K-1. (PFor b)
The integer solution obtained in step)
Further scheme is：
Maximal possibility estimation (MLE) model is built, including：
The probability density function that I sets up noise (includes but is not limited to Gaussian distributed)；
The II calculating actual times of arrival are expressed as d0+ε；
III object functions minimize for the quadratic sum of noise；
Noise signal to noise ratio (SNR) substantial scope that IV considers is 0-100dB.
Further scheme is：
Relaxation step in step b), positive semidefinite relaxation and quadric cone relaxation, obtain the maximal possibility estimation after following relaxation
The convex optimization problem model (2) of problem, i.e. MIXED INTEGER
Wherein first constraint is quadric cone relaxation, and second, the 3rd, the 4th and the 5th constraints is partly just
Fixed relaxation, is relaxed by positive semidefinite and the convex optimization problem of a MIXED INTEGER-positive semidefinite-quadric cone was obtained after being relaxed with quadric cone
Model (2).
Further scheme is：
Taylor's relaxation is comprised the following steps that：
By to carry out Taylor's single order relaxation of second constraints in primal problem model (1), second order relaxation or
High-order relaxes, so that a loose single order, second order or high-order inequality are obtained, so that it is whole that problem is converted into a mixing
The convex optimization problem of number-quadric cone.
Further scheme is：
Positive semidefinite matrix (SDP) constraints after the new relaxation of addition, be specially：
Positive semidefinite matrix D after being relaxed to positive semidefinite adds constraints, to ensure diagonal element element in positive semidefinite matrix
Di k,kNumerical value is as close possible to (Di k,L+1)2, so that it is guaranteed that positive semidefinite relaxation is with former problem equivalent, the method used includes but not limited
In addition constraints：
Further scheme is：
Outer approximation approximate data submodel (3), its model is including but not limited to feature as follows：
<Di, Ddual { iternum }i>≥0
<Xk, Xdual { iternum }k>≥0
Outer approximation approximate data submodel (3i)
ZK, i>=0, Pi∈ Π,
Outer approximation approximate data submodel (3ii)
Further scheme is：
Convex row optimization problem model (4), its model is including but not limited to feature as follows：
Convex row optimization problem model (4)
The model is a continuous-convex optimization problem of positive semidefinite-quadric cone, and convex optimized algorithm can be used to obtain optimal
Solution, wherein first constraints represents that the integer solution that will be obtained brings our TOA data into, so that the 0-1 integer variables that disappear.
Further scheme is：
The corresponding positive semidefinite slack variable D of the true arrival time d of D_dual dual optimal solution D_dual, and target location
The corresponding positive semidefinite slack variable X of known variables x dual optimal solution X_dual, can be convex excellent with following characteristics by calculating
Change problem is obtained：
The corresponding dual problem model of relaxation problem
Wherein second and the 3rd constraint distribution represent D_dual and X_dual.
Two more specifically embodiments are enumerated below according to actual conditions to illustrate.
Embodiment 1
Base station：
S=
| 40 | 40 | -40 | -40 | 40 | 0 | -40 | 0 |
| 40 | -40 | 40 | -40 | 0 | 40 | 0 | 40 |
Target to be positioned：
X=
| 10 | -20 | 30 |
| -10 | -25 | 20 |
After 100 Monte Carlo simulations, as shown in Figure 2, Fig. 2 shows 3 targets 8 to the target location scatter diagram of calculating
The special Monte Carlo Simulation of Ions Inside result figure of the door of individual base station.
Embodiment 2
Base station：
S=
| 40 | 40 | -40 | -40 |
| 40 | -40 | 40 | -40 |
Target to be positioned：
X=
| -20 | 0 | 20 | -20 | 0 | 20 | -20 | 0 | 20 |
| 20 | 20 | 20 | 0 | 0 | 0 | -20 | -20 | -20 |
Signal to noise ratio
| SNR(dB) | 10 | 20 | 30 | 40 |
After 100 Monte Carlo simulations, as shown in Figure 3, Fig. 3 shows 9 targets 4 to the target location scatter diagram of calculating
The special Monte Carlo Simulation of Ions Inside result of the door of individual base station.
The multiple target of table 1 (9) positioning precision and computation complexity
All numerical experiments are completed by notebook computer, can be solved as the algorithm described by accompanying drawing 2,3 understands this patent
Certainly TOA-MSL orientation problems, when signal to noise ratio is more than or equal to 20dB, can obtain good numerical result.Table 1 shows, uses this
Algorithm described by patent, which solves multi-objective problem, only to be needed less than 9 seconds, was significantly less than known Multi-target position algorithm,
Noise can be close to CRLB precision when smaller.
Although reference be made herein to invention has been described for explanatory embodiment of the invention, and above-described embodiment is only this hair
Bright preferably embodiment, embodiments of the present invention are simultaneously not restricted to the described embodiments, it should be appreciated that people in the art
Member can be designed that a lot of other modification and embodiment, and these modifications and embodiment will fall in principle disclosed in the present application
Within scope and spirit.
Claims (13)
1. a kind of approximate convex optimized algorithm of Multi-target position outer approximation based on arrival time, including relaxed by positive semidefinite,
After quadric cone relaxation, Taylor's relaxation and addition SDP Condition of Strong Constraint, a mixing is obtained according to maximal possibility estimation MLE models
The convex optimization problem of integer, accurate multiple target coordinate is obtained according to outer approximation approximate data, it is characterised in that specifically include with
Lower step：
A) defined by TOA, the primal problem model of a MIXED INTEGER, non-convex is constructed using maximal possibility estimation；
B) relaxed by using positive semidefinite, Taylor relaxation and quadric cone relaxation etc. relaxation method and addition SDP constraintss, will
Former problem relaxation is the convex optimization problem model of a MIXED INTEGER；
C) solution of an integer part is obtained by solving outer approximation approximate data submodelP, corresponding optimal function value conduct
The lower bound LB of whole algorithm；
D) fixed integer solution part, b) the convex optimization problem model of corresponding MIXED INTEGER be changed into a convex row optimization problem mould
Type, minimum in continuous part solution, and corresponding functional value, respective function value is obtained by solving the convex row optimization problem model
It is worth the upper bound UB for whole algorithm；
E) judge solution that convex row optimization problem model obtains whether be former problem optimal solution：
ⅰ.If UB>LB+constant, d) in continuous solution and the integer solution corresponding with him be not former problem optimal solution,
The constraints of outer approximation approximate data submodel in iteration upgrading b), it is ensured that the integer solution occurred will not be again
Occur, continue executing with b)-d) until obtaining optimal solution or iterations more than the default upper limit；
ⅱ.If UB<=LB+constant, d) in continuous solution and the integer solution corresponding with him be the optimal of former problem
Solution, shuts down procedure.
2. the approximate convex optimized algorithm of Multi-target position outer approximation based on arrival time, its feature according to claim 1
It is：
Step a) includes：
I .TOA is defined, it is assumed that have its one L × K matrix of correspondence of K target x to be positioned, M base station s one L × M square of correspondence
Battle array, the wherein location matrix s of base station is, it is known that arrival time corresponding K × Metzler matrix d is, it is known that the location matrix x of target is unknown；Root
Understand that the arrival time of k-th of target to i-th of base station is as follows according to TOA definition：
<mrow>
<msub>
<mi>d</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>c</mi>
</mfrac>
<mo>|</mo>
<mo>|</mo>
<msub>
<mi>x</mi>
<mrow>
<mo>:</mo>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>-</mo>
<msub>
<mi>s</mi>
<mrow>
<mo>:</mo>
<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
<mo>|</mo>
<mo>|</mo>
<mo>+</mo>
<msub>
<mi>&epsiv;</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
</mrow>
Wherein, last is noise, without loss of generality, it will be assumed that each noise Gaussian distributed, and independently
With distribution；In Multi-target position, it is assumed that can not recognize the signal that he obtains comes from which target, table for each base station
It is shown asIn order to recognize each target, we are that each base station introduces a permutation matrix to recognize each mesh
Mark, for i-th of base station, recognizes the TOA values from k-th of target, method is as follows：
<mrow>
<msubsup>
<mi>P</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mo>:</mo>
</mrow>
<mi>i</mi>
</msubsup>
<mover>
<msub>
<mi>d</mi>
<mrow>
<mo>:</mo>
<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
<mo>^</mo>
</mover>
</mrow>
II goes to minimize the absolute value of noise, following Multi-target position model is constructed based on this by maximal possibility estimation,
That is primal problem model：
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<munder>
<mrow>
<mi>m</mi>
<mi>i</mi>
<mi>n</mi>
</mrow>
<mrow>
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<mi>P</mi>
<mo>,</mo>
<msup>
<mi>d</mi>
<mn>0</mn>
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<mi>x</mi>
</mrow>
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<mrow>
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<mn>1</mn>
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<mi>K</mi>
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<mn>1</mn>
</mrow>
<mi>M</mi>
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<mi>Z</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
</mrow>
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<mtable>
<mtr>
<mtd>
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<mi>t</mi>
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</mtd>
</mtr>
</mtable>
</mfenced>
1
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<mi>d</mi>
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<mo>,</mo>
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</mrow>
<mn>0</mn>
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<mi>c</mi>
</mfrac>
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<mi>k</mi>
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<msub>
<mi>s</mi>
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<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
<mo>|</mo>
<mo>|</mo>
<mo>,</mo>
</mrow>
ZK, i>=0, Pi∈ π,
<mrow>
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<mi>i</mi>
<mo>=</mo>
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<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>M</mi>
<mo>,</mo>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>K</mi>
<mo>.</mo>
</mrow>
Wherein, PiRepresent the corresponding permutation matrix in i-th of base station；The TOA data received for i-th of base station；Zk,iRepresent kth
Square of the actual TOA data of individual target and i-th base station and the difference of estimated data；For k-th of target to be calculated with
The true TOA values of i-th of base station；Second constraint is TOA physical definition.
3. the approximate convex optimized algorithm of Multi-target position outer approximation based on arrival time, its feature according to claim 1
It is：
Step b) includes：
I calculates the position of multiple targets according to the TOA data and the positional information of base station；
II carries out including but is not limited to positive semidefinite relaxation, quadric cone relaxation, Taylor's relaxation and addition SDP constraints to primal problem
Condition with cause former problem be changed into convex problem and reduction computation complexity.
4. the approximate convex optimized algorithm of Multi-target position outer approximation based on arrival time, its feature according to claim 1
It is：
The step c) includes：
The positive semi-definite constraints of SDP constraints is fallen in I relaxations, obtains outer approximation approximate data submodel；
II outer approximation approximate data submodels are a convex optimization problems of MIXED INTEGER-quadric cone；
Outer approximation approximate data submodel after III relaxations, can use including but not limited to branch-bound algorithm, interior point
Method, fixed step size gradient descent method, variable step-length gradient drop method and Newton method obtain the optimal solution and optimal function of subproblem
Value；
IV preserves the integer part of optimal solution, and optimal function value is defined as the lower bound LB of former problem, into step c).
5. the approximate convex optimized algorithm of Multi-target position outer approximation based on arrival time, its feature according to claim 1
It is：
Step d) includes：
Behind I fixed integers part, the convex optimization problem model of former MIXED INTEGER become in order to one continuously with positive semidefinite constraint and
The convex optimization problem of continuously differentiable point of quadric cone constraint；
The II problems use the ripe convex optimization of interior point method, the gradient descent method of fixed step size or change step and Newton method
Algorithm obtains the optimal solution and optimal function value of the subproblem；
III optimal function values are designated as the upper bound UB of former problem.
6. the approximate convex optimized algorithm of Multi-target position outer approximation based on arrival time, its feature according to claim 1
It is：
Step e) includes：
Ensure that the unduplicated method of integer solution includes but is not limited to：
The dual problem of convex row optimization problem model, obtains the location variable x corresponding antithesis of slack variable X during I is calculated d)
Optimal solution X_dual and the corresponding slack variable D of true arrival time d dual optimal solution D_dual, add in subproblem b)
Plus about<X,X_dual>>=0, i.e. Kronecker product；<D,D_dual>>=0.
II directly adds constraint in outer approximation approximate data submodel b)<P,P><=M × K-1. is whereinPFor in b) step
Obtained integer solution.
7. the approximate convex optimized algorithm of Multi-target position outer approximation based on arrival time, its feature according to claim 2
It is：
By maximal possibility estimation, maximal possibility estimation model is built, including：
I sets up the probability density function of noise, including but not limited to Gaussian distributed
The II calculating actual times of arrival are expressed as d0+ε；
III object functions minimize for the quadratic sum of noise；
The noise SNR ranges that IV considers are 0-100dB.
8. the approximate convex optimized algorithm of Multi-target position outer approximation based on arrival time, its feature according to claim 1
It is：
Relaxation step in step b), positive semidefinite relaxation and quadric cone relaxation, obtain the maximal possibility estimation problem after following relaxation,
That is the convex optimization problem model of MIXED INTEGER
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<mo>,</mo>
<mi>X</mi>
<mo>,</mo>
<mi>D</mi>
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<mi>P</mi>
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<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
</mrow>
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<mtable>
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<mrow>
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<mi>t</mi>
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</mtd>
<mtd>
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<mo>|</mo>
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<mo>,</mo>
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<mi>i</mi>
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<mi>i</mi>
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</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mrow>
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<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
<mi>i</mi>
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<mi>t</mi>
<mi>r</mi>
<mi>a</mi>
<mi>c</mi>
<mi>e</mi>
<mrow>
<mo>(</mo>
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<mover>
<mi>S</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msup>
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<mi>X</mi>
<mi>k</mi>
</msup>
<mo>)</mo>
</mrow>
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</mrow>
<mrow>
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<mi>D</mi>
<mrow>
<mn>1</mn>
<mo>:</mo>
<mi>K</mi>
<mo>,</mo>
<mi>K</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mi>i</mi>
</msubsup>
<mo>=</mo>
<msup>
<mrow>
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<msubsup>
<mi>D</mi>
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<mi>i</mi>
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Wherein first constraint is quadric cone relaxation, and second, the 3rd, the 4th and the 5th constraints is positive semidefinite pine
Relax, relaxed by positive semidefinite and the convex optimization problem mould of a MIXED INTEGER-positive semidefinite-quadric cone was obtained after being relaxed with quadric cone
Type.
9. the approximate convex optimized algorithm of Multi-target position outer approximation based on arrival time, its feature according to claim 1
It is：
Taylor's relaxation is comprised the following steps that：
By to carry out Taylor's single order relaxation of second constraints, second order relaxation or high-order pine in primal problem model
Relax, so as to obtain a loose single order, second order or high-order inequality, so that problem is converted into a MIXED INTEGER-secondary
The convex optimization problem of cone.
10. the approximate convex optimized algorithm of Multi-target position outer approximation based on arrival time, its feature according to claim 1
It is：
Positive semidefinite matrix constraints after the new relaxation of addition, be specially：
Positive semidefinite matrix after being relaxed to positive semidefinite adds constraints, to ensure diagonal element element D in positive semidefinite matrixi k,kNumber
Value is as close possible to (Di k,L+1)2, so that it is guaranteed that positive semidefinite relaxation with not relaxation former problem equivalent, the method used include but
It is not limited to add constraints：
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<mi>k</mi>
</mrow>
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3
11. the approximate convex optimized algorithm of Multi-target position outer approximation based on arrival time, its feature according to claim 1
It is：
The outer approximation approximate data submodel, its model is including but not limited to feature as follows：
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<mi>m</mi>
<mi>i</mi>
<mi>n</mi>
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<mo>,</mo>
<mi>i</mi>
</mrow>
<mn>0</mn>
</msubsup>
<mi>&delta;</mi>
</mrow>
<mrow>
<msubsup>
<mi>D</mi>
<mrow>
<mn>1</mn>
<mo>:</mo>
<mi>K</mi>
<mo>,</mo>
<mi>K</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mi>i</mi>
</msubsup>
<mo>=</mo>
<msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>D</mi>
<mrow>
<mi>K</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mn>1</mn>
<mo>:</mo>
<mi>K</mi>
</mrow>
<mi>i</mi>
</msubsup>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
<mo>=</mo>
<msubsup>
<mi>d</mi>
<mrow>
<mo>:</mo>
<mo>,</mo>
<mi>i</mi>
</mrow>
<mn>0</mn>
</msubsup>
</mrow>
<Di, Ddual { iternum }i>≥0
<Xk, Xdual { iternum }k>≥0
<mrow>
<msubsup>
<mi>D</mi>
<mrow>
<mi>K</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mi>K</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mi>i</mi>
</msubsup>
<mo>=</mo>
<msubsup>
<mi>X</mi>
<mrow>
<mi>L</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mi>L</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mi>k</mi>
</msubsup>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow>
<mo>&ForAll;</mo>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>M</mi>
</mrow>
<mrow>
<mo>&ForAll;</mo>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>K</mi>
</mrow>
With
<mrow>
<munder>
<mrow>
<mi>m</mi>
<mi>i</mi>
<mi>n</mi>
</mrow>
<mrow>
<mi>Z</mi>
<mo>,</mo>
<mi>X</mi>
<mo>,</mo>
<mi>D</mi>
<mo>,</mo>
<mi>P</mi>
<mo>,</mo>
<msup>
<mi>d</mi>
<mn>0</mn>
</msup>
</mrow>
</munder>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>K</mi>
</munderover>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>M</mi>
</munderover>
<msub>
<mi>Z</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mfenced open = "" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>s</mi>
<mo>.</mo>
<mi>t</mi>
<mo>.</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>|</mo>
<mo>|</mo>
<msubsup>
<mi>P</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mo>:</mo>
</mrow>
<mi>i</mi>
</msubsup>
<msub>
<mover>
<mi>d</mi>
<mo>^</mo>
</mover>
<mrow>
<mo>:</mo>
<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
<mo>-</mo>
<msubsup>
<mi>d</mi>
<mrow>
<mo>:</mo>
<mo>,</mo>
<mi>i</mi>
</mrow>
<mn>0</mn>
</msubsup>
<mo>|</mo>
<mo>|</mo>
<mo>&le;</mo>
<msub>
<mi>Z</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mrow>
<msubsup>
<mi>D</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
<mi>i</mi>
</msubsup>
<mo>=</mo>
<mi>t</mi>
<mi>r</mi>
<mi>a</mi>
<mi>c</mi>
<mi>e</mi>
<mrow>
<mo>(</mo>
<msup>
<mover>
<mi>S</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msup>
<msup>
<mi>X</mi>
<mi>k</mi>
</msup>
<mo>)</mo>
</mrow>
<mo>,</mo>
</mrow>
<mrow>
<msubsup>
<mi>D</mi>
<mrow>
<mn>1</mn>
<mo>:</mo>
<mi>K</mi>
<mo>,</mo>
<mi>K</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mi>i</mi>
</msubsup>
<mo>=</mo>
<msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>D</mi>
<mrow>
<mi>K</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mn>1</mn>
<mo>:</mo>
<mi>K</mi>
</mrow>
<mi>i</mi>
</msubsup>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
<mo>=</mo>
<msubsup>
<mi>d</mi>
<mrow>
<mo>:</mo>
<mo>,</mo>
<mi>i</mi>
</mrow>
<mn>0</mn>
</msubsup>
<mo>,</mo>
</mrow>
<mrow>
<mo><</mo>
<mi>P</mi>
<mo>,</mo>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mo>></mo>
<mo>&le;</mo>
<mi>M</mi>
<mo>&times;</mo>
<mi>K</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
ZK, i>=0, Pi∈ Π,
<mrow>
<msubsup>
<mi>D</mi>
<mrow>
<mi>K</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mi>K</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mi>i</mi>
</msubsup>
<mo>=</mo>
<msubsup>
<mi>X</mi>
<mrow>
<mi>L</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mi>L</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mi>k</mi>
</msubsup>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
</mrow>
<mrow>
<mo>&ForAll;</mo>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>M</mi>
<mo>,</mo>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>K</mi>
<mo>,</mo>
</mrow>
<mrow>
<mo>&ForAll;</mo>
<mi> </mi>
<mi>i</mi>
<mi>t</mi>
<mi>e</mi>
<mi>r</mi>
<mi>n</mi>
<mi>u</mi>
<mi>m</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>i</mi>
<mi>t</mi>
<mi>e</mi>
<mi>r</mi>
<mi> </mi>
<mi>E</mi>
<mn>..</mn>
</mrow>
12. the approximate convex optimized algorithm of Multi-target position outer approximation based on arrival time, its feature according to claim 1
It is：
Convex row optimization problem model, its model is including but not limited to feature as follows：
<mrow>
<munder>
<mrow>
<mi>m</mi>
<mi>i</mi>
<mi>n</mi>
</mrow>
<mrow>
<mi>Z</mi>
<mo>,</mo>
<mi>D</mi>
<mo>,</mo>
<msup>
<mi>d</mi>
<mn>0</mn>
</msup>
<mo>,</mo>
<mi>X</mi>
</mrow>
</munder>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>K</mi>
</munderover>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>M</mi>
</munderover>
<msub>
<mi>Z</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi>s</mi>
<mo>.</mo>
<mi>t</mi>
<mo>.</mo>
<msub>
<mover>
<mi>d</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mo>:</mo>
<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<mo>|</mo>
<mo>|</mo>
<msubsup>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mo>:</mo>
</mrow>
<mi>i</mi>
</msubsup>
<msub>
<mover>
<mi>d</mi>
<mo>^</mo>
</mover>
<mrow>
<mo>:</mo>
<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
<mo>|</mo>
<mo>|</mo>
</mrow>
<mrow>
<mo>|</mo>
<mo>|</mo>
<msub>
<mover>
<mi>d</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
<mo>-</mo>
<msubsup>
<mi>d</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
<mn>0</mn>
</msubsup>
<mo>|</mo>
<mo>|</mo>
<mo>&le;</mo>
<msub>
<mi>Z</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<msubsup>
<mi>D</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
<mi>i</mi>
</msubsup>
<mo>=</mo>
<mi>t</mi>
<mi>r</mi>
<mi>a</mi>
<mi>c</mi>
<mi>e</mi>
<mrow>
<mo>(</mo>
<msup>
<mover>
<mi>S</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msup>
<msup>
<mi>X</mi>
<mi>k</mi>
</msup>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msubsup>
<mi>P</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mo>:</mo>
</mrow>
<mi>i</mi>
</msubsup>
<mi>d</mi>
<mi>i</mi>
<mi>a</mi>
<mi>g</mi>
<mrow>
<mo>(</mo>
<msub>
<mover>
<mi>d</mi>
<mo>^</mo>
</mover>
<mrow>
<mo>:</mo>
<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
<msubsup>
<mover>
<mi>d</mi>
<mo>^</mo>
</mover>
<mrow>
<mo>:</mo>
<mo>,</mo>
<mi>i</mi>
</mrow>
<mi>T</mi>
</msubsup>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msup>
<mi>&delta;</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<mn>2</mn>
<msubsup>
<mi>P</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mo>:</mo>
</mrow>
<mi>i</mi>
</msubsup>
<msub>
<mover>
<mi>d</mi>
<mo>^</mo>
</mover>
<mrow>
<mo>:</mo>
<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
<mi>&delta;</mi>
<mo>&le;</mo>
<msubsup>
<mi>D</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
<mi>i</mi>
</msubsup>
</mrow>
4
<mrow>
<msubsup>
<mi>D</mi>
<mrow>
<mi>k</mi>
<mi>k</mi>
</mrow>
<mi>i</mi>
</msubsup>
<mo>&le;</mo>
<msubsup>
<mi>P</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mo>:</mo>
</mrow>
<mi>i</mi>
</msubsup>
<mi>d</mi>
<mi>i</mi>
<mi>a</mi>
<mi>g</mi>
<mrow>
<mo>(</mo>
<msub>
<mover>
<mi>d</mi>
<mo>^</mo>
</mover>
<mrow>
<mo>:</mo>
<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
<msubsup>
<mover>
<mi>d</mi>
<mo>^</mo>
</mover>
<mrow>
<mo>:</mo>
<mo>,</mo>
<mi>i</mi>
</mrow>
<mi>T</mi>
</msubsup>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>3</mn>
<msup>
<mi>&delta;</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<mn>2</mn>
<msubsup>
<mi>d</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
<mn>0</mn>
</msubsup>
<mi>&delta;</mi>
</mrow>
<mrow>
<msubsup>
<mi>D</mi>
<mrow>
<mn>1</mn>
<mo>:</mo>
<mi>K</mi>
<mo>,</mo>
<mi>K</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mi>i</mi>
</msubsup>
<mo>=</mo>
<msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>D</mi>
<mrow>
<mi>K</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mn>1</mn>
<mo>:</mo>
<mi>K</mi>
</mrow>
<mi>i</mi>
</msubsup>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
<mo>=</mo>
<msubsup>
<mi>d</mi>
<mrow>
<mo>:</mo>
<mo>,</mo>
<mi>i</mi>
</mrow>
<mn>0</mn>
</msubsup>
</mrow>
<mrow>
<msubsup>
<mi>D</mi>
<mrow>
<mi>K</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mi>K</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mi>i</mi>
</msubsup>
<mo>=</mo>
<msubsup>
<mi>X</mi>
<mrow>
<mi>L</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mi>L</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mi>k</mi>
</msubsup>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow>
<mo>&ForAll;</mo>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>M</mi>
</mrow>
<mrow>
<mo>&ForAll;</mo>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>K</mi>
</mrow>
The model is a continuous-convex optimization problem of positive semidefinite-quadric cone, convex optimized algorithm can be used to obtain optimal solution, its
In first constraints represent that the integer solution that will be obtained brings our TOA data into so that the 0-1 integer variables that disappear.
13. the approximate convex optimized algorithm of Multi-target position outer approximation based on arrival time, its feature according to claim 6
It is：
The corresponding positive semidefinite slack variable D of the true arrival time d of D_dual dual optimal solution D_dual, and target location are unknown
The corresponding positive semidefinite slack variable X of variable x dual optimal solution X_dual, by calculating the relaxation problem pair with following characteristics
The dual problem model answered is obtained：
<mrow>
<munder>
<mi>max</mi>
<mrow>
<mi>&alpha;</mi>
<mo>,</mo>
<mi>&gamma;</mi>
<mo>,</mo>
<mi>&eta;</mi>
<mo>,</mo>
<mi>&beta;</mi>
<mo>,</mo>
<mi>&delta;</mi>
</mrow>
</munder>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>K</mi>
</munderover>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>M</mi>
</munderover>
<msubsup>
<mi>&alpha;</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
<mrow>
<mi>d</mi>
<mi>T</mi>
</mrow>
</msubsup>
<msub>
<mover>
<mi>d</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
<mo>-</mo>
<munder>
<mo>&Sigma;</mo>
<mi>i</mi>
</munder>
<msubsup>
<mi>&gamma;</mi>
<mi>i</mi>
<mi>D</mi>
</msubsup>
<mo>-</mo>
<munder>
<mo>&Sigma;</mo>
<mi>k</mi>
</munder>
<msubsup>
<mi>&gamma;</mi>
<mi>k</mi>
<mi>X</mi>
</msubsup>
</mrow>
<mfenced open = "" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>s</mi>
<mo>.</mo>
<mi>t</mi>
<mo>.</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>1</mn>
<mo>-</mo>
<msubsup>
<mi>&alpha;</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
<mi>Z</mi>
</msubsup>
<mo>&GreaterEqual;</mo>
<mn>0</mn>
<mo>,</mo>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mrow>
<msubsup>
<mi>&alpha;</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
<mrow>
<mi>d</mi>
<mi>T</mi>
</mrow>
</msubsup>
<mo>-</mo>
<msubsup>
<mi>&eta;</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
<mrow>
<mn>1</mn>
<mi>T</mi>
</mrow>
</msubsup>
<mo>-</mo>
<msubsup>
<mi>&eta;</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
<mrow>
<mn>2</mn>
<mi>T</mi>
</mrow>
</msubsup>
<mo>&GreaterEqual;</mo>
<mn>0</mn>
<mo>,</mo>
</mrow>
<mrow>
<mo>|</mo>
<mo>|</mo>
<msubsup>
<mi>&alpha;</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
<mi>d</mi>
</msubsup>
<mo>|</mo>
<mo>|</mo>
<mo>&le;</mo>
<msubsup>
<mi>&alpha;</mi>
<mrow>
<mi>k</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
<mi>Z</mi>
</msubsup>
<mo>,</mo>
</mrow>
<mrow>
<msubsup>
<mi>&gamma;</mi>
<mi>i</mi>
<mi>D</mi>
</msubsup>
<mo>,</mo>
<msubsup>
<mi>&gamma;</mi>
<mi>k</mi>
<mi>X</mi>
</msubsup>
<mo>&Element;</mo>
<mi>R</mi>
<mo>,</mo>
<msub>
<mi>&delta;</mi>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>&GreaterEqual;</mo>
<mn>0</mn>
<mo>,</mo>
</mrow>
<mrow>
<mo>&ForAll;</mo>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>K</mi>
<mo>,</mo>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>M</mi>
<mo>.</mo>
</mrow>
Wherein second and the 3rd constraint distribution represent D_dual and X_dual.
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