Variational-like inequalities with set-valued mappings are very useful in economics and nonsmooth optimization problems. In this paper, we study the existence of solutions and the formulation of solution methods for v...
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Variational-like inequalities with set-valued mappings are very useful in economics and nonsmooth optimization problems. In this paper, we study the existence of solutions and the formulation of solution methods for vector variational-like inequalities (VVLI) with set-valued mappings. We introduce gap functions and establish necessary and sufficient conditions for the existence of a solution of the VVLI. We investigate the existence of a solution for the generalized VVLI with a set-valued mapping by exploiting the existence of a solution of the VVLI with a single-valued function and a continuous selection theorem.
We propose primal - dual path- following Mehrotra- type predictor corrector methods for solving convex quadratic semidefinite programming (QSDP) problems of the form: min(X){1/2X center dot Q(X) + C center dot X : A( ...
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We propose primal - dual path- following Mehrotra- type predictor corrector methods for solving convex quadratic semidefinite programming (QSDP) problems of the form: min(X){1/2X center dot Q(X) + C center dot X : A( X) = b, X >= 0}, where Q is a self- adjoint positive semidefinite linear operator on S (n), b is an element of R-m, and A is a linear map from S n to Rm. At each interior- point iteration, the search direction is computed from a dense symmetric indefinite linear system ( called the augmented equation) of dimension m+ n( n+ 1)/ 2. Such linear systems are typically very large and can only be solved by iterative methods. We propose three classes of preconditioners for the augmented equation, and show that the corresponding preconditioned matrices have favorable asymptotic eigenvalue distributions for fast convergence under suitable nondegeneracy assumptions. Numerical experiments on a variety of QSDPs with n up to 1600 are performed and the computational results show that our methods are efficient and robust.
We consider two notions for the representations of convex cones G-representation and lifted-G-representation. The former represents a convex cone as a slice of another;the latter allows in addition, the usage of auxil...
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We consider two notions for the representations of convex cones G-representation and lifted-G-representation. The former represents a convex cone as a slice of another;the latter allows in addition, the usage of auxiliary variables in the representation. We first study the basic properties of these representations. We show that some basic properties of convex cones are invariant under one notion of representation but not the other. In particular, we prove that lifted-G-representation is closed under duality when the representing cone is self-dual. We also prove that strict complementarity of a convex optimization problem in conic form is preserved under G-representations. Then we move to study efficiency measures for representations. We evaluate the representations of homogeneous convex cones based on the "smoothness" of the transformations mapping the central path of the representation to the central path of the represented optimization problem.
We consider two notions for the representations of convex cones G-representation and lifted-G-representation. The former represents a convex cone as a slice of another;the latter allows in addition, the usage of auxil...
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We consider two notions for the representations of convex cones G-representation and lifted-G-representation. The former represents a convex cone as a slice of another;the latter allows in addition, the usage of auxiliary variables in the representation. We first study the basic properties of these representations. We show that some basic properties of convex cones are invariant under one notion of representation but not the other. In particular, we prove that lifted-G-representation is closed under duality when the representing cone is self-dual. We also prove that strict complementarity of a convex optimization problem in conic form is preserved under G-representations. Then we move to study efficiency measures for representations. We evaluate the representations of homogeneous convex cones based on the "smoothness" of the transformations mapping the central path of the representation to the central path of the represented optimization problem.
Support vector machines (SVMs) have been successfully applied to classification problems. Practical issues involve bow to determine the right type and suitable hyperparameters of kernel functions. Recently, multiple-k...
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ISBN:
(纸本)9781424420957
Support vector machines (SVMs) have been successfully applied to classification problems. Practical issues involve bow to determine the right type and suitable hyperparameters of kernel functions. Recently, multiple-kernel learning (MKL) algorithms are developed to handle these issues by combining different kernels. The weight with each kernel in the combination is obtained through learning. One of the most popular methods is to learn the weights with semidefinite programming (SDP). However, the amount of time and space required by this method is demanding. In this study. we reformulate the SDP problem to reduce the time and space requirements. Strategies for reducing the search space in solving the SDP problem are introduced. Experimental results obtained from running on synthetic datasets and benchmark datasets of UCI and Stating show that the proposed approach improves the efficiency of the SDP method without degrading the performance.
There are various conditions on the CS matrix for unique and stable recovery. These include universality, or spark, and UUP. Furthermore, quantitative bounds on the stability depend on related properties of the CS mat...
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ISBN:
(纸本)9781424414833
There are various conditions on the CS matrix for unique and stable recovery. These include universality, or spark, and UUP. Furthermore, quantitative bounds on the stability depend on related properties of the CS matrix. The construction of good CS matrices - satisfying the various properties - is key to successful practical applications of compressive sensing. Unfortunately, verifying the satisfiability of any of these properties for a given CS matrix involves infeasible combinatorial search. Our methods use l(1) and semidefinite relaxation into a convex problem. Given a set of candidate CS matrices, our approach provides tools for the selection of good CS matrices with verified and quantitatively favorable performance.
We investigate the problem of robust joint transmitter and receiver power allocation for uplink multi-input multi-output (MIMO) transmissions. The channel model is assumed to have Rayleigh flat fading. The objective o...
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ISBN:
(纸本)9781424423354
We investigate the problem of robust joint transmitter and receiver power allocation for uplink multi-input multi-output (MIMO) transmissions. The channel model is assumed to have Rayleigh flat fading. The objective of power allocation is to minimize the total MSE each of which has limited transmit power. The problem is formulated as a non-linear optimization problem. In addition, we investigate robust joint transmitter and receiver power allocation with imperfect channel state information (CSI). The CSI error is assumed to be unknown but bounded by a constant. This problem is formulated as a semidefinite programming (SDP) problem with bilinear matrix inequality (BMI) constraints. Numerical results indicate that, with imperfect CSI, the BER performance improvement obtained by taking robustness into account in the joint power allocation process.
We consider the positive semidefinite (psd) matrices with binary entries. We give a characterisation of such matrices, along with a graphical representation. We then move on to consider the associated integer polytope...
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ISBN:
(纸本)9783540688860
We consider the positive semidefinite (psd) matrices with binary entries. We give a characterisation of such matrices, along with a graphical representation. We then move on to consider the associated integer polytopes. Several important and well-known integer polytopes - the cut, boolean quadric, multicut and clique partitioning polytopes - are shown to arise as projections of binary psd polytopes. Finally, we present various valid inequalities for binary psd polytopes, and show how they relate to inequalities known for the simpler polytopes mentioned. Along the way, we answer an open question in the literature on the max-cut problem, by showing that the so-called k-gonal inequalities define a polytope.
We propose primal - dual path- following Mehrotra- type predictor corrector methods for solving convex quadratic semidefinite programming (QSDP) problems of the form: min(X){1/2X center dot Q(X) + C center dot X : A( ...
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We propose primal - dual path- following Mehrotra- type predictor corrector methods for solving convex quadratic semidefinite programming (QSDP) problems of the form: min(X){1/2X center dot Q(X) + C center dot X : A( X) = b, X >= 0}, where Q is a self- adjoint positive semidefinite linear operator on S (n), b is an element of R-m, and A is a linear map from S n to Rm. At each interior- point iteration, the search direction is computed from a dense symmetric indefinite linear system ( called the augmented equation) of dimension m+ n( n+ 1)/ 2. Such linear systems are typically very large and can only be solved by iterative methods. We propose three classes of preconditioners for the augmented equation, and show that the corresponding preconditioned matrices have favorable asymptotic eigenvalue distributions for fast convergence under suitable nondegeneracy assumptions. Numerical experiments on a variety of QSDPs with n up to 1600 are performed and the computational results show that our methods are efficient and robust.
semidefinite programming(SDP) is one of the strongest algorithmic techniques used in the design of approximation algorithms. In recent years, Unique Games Conjecture(UGC) has proved to be intimately connected to the l...
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ISBN:
(纸本)9781605580470
semidefinite programming(SDP) is one of the strongest algorithmic techniques used in the design of approximation algorithms. In recent years, Unique Games Conjecture(UGC) has proved to be intimately connected to the limitations of semidefinite programming. Making this connection precise, we show the following result : If UGC is true, then for every constraint;satisfaction problem(CSP) the best approximation ratio is given by a certain simple SDP. Specifically, we show a, generic conversion from SDP integrality gaps to UGC hardness results or every CSP. This result holds both for maximization and minimization problems over arbitrary finite domains. Using this connection between integrality gaps and hardness results we obtain a generic polynomial-time algorithm for all CSPs. Assuming the Unique Carries Conjecture, this algorithm achieves the optimal approximation ratio for every CSP. Unconditionally, for all 2-CSPs the algorithm achieves ail approximation ratio equal to the integrality gap of a natural SDP used in literature. Further the algorithm achieves at least as good an approximation ratio as the best known algorithms for several problems like MaxCut, Max2Sat, MaxDiCut and Unique Games.
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