In semidefinite programming, one minimizes a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but conve...
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In semidefinite programming, one minimizes a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but convex, so semidefinite programs are convex optimization problems. semidefinite programming unifies several standard problems (e.g., linear and quadratic programming) and finds many applications in engineering and combinatorial optimization. Although semidefinite programs are much more general than linear programs, they are not much harder to solve. Most interior-point methods for linear programming have been generalized to semidefinite programs. As in linear programming, these methods have polynomial worst-case complexity and perform very well in practice. This paper gives a survey of the theory and applications of semidefinite programs and an introduction to primal-dual interior-point methods for their solution.
We show that the Christensen-Sinclair factorization theorem, when the underlying Hilbert spaces are finite dimensional, is an instance of strong duality of semidefinite programming. This gives an elementary proof of t...
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We show that the Christensen-Sinclair factorization theorem, when the underlying Hilbert spaces are finite dimensional, is an instance of strong duality of semidefinite programming. This gives an elementary proof of the result and also provides an efficient algorithm to compute the Christensen-Sinclair factorization. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://
In many applications, solutions of convex optimization problems are updated on-line, as functions of time. In this paper, we consider parametric semidefinite programs, which are linear optimization problems in the sem...
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In many applications, solutions of convex optimization problems are updated on-line, as functions of time. In this paper, we consider parametric semidefinite programs, which are linear optimization problems in the semidefinite cone whose coefficients (input data) depend on a time parameter. We are interested in the geometry of the solution (output data) trajectory, defined as the set of solutions depending on the parameter. We propose an exhaustive description of the geometry of the solution trajectory. As our main result, we show that only six distinct behaviors can be observed at a neighborhood of a given point along the solution trajectory. Each possible behavior is then illustrated by an example.
The minimum sum-of-squares clustering (MSSC), or k-means type clustering, has been recently extended to exploit prior knowledge on the cardinality of each cluster. Such knowledge is used to increase performance as wel...
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The minimum sum-of-squares clustering (MSSC), or k-means type clustering, has been recently extended to exploit prior knowledge on the cardinality of each cluster. Such knowledge is used to increase performance as well as solution quality. In this paper, we propose a global optimization approach based on the branch-and-cut technique to solve the cardinality-constrained MSSC. For the lower bound routine, we use the semidefinite programming (SDP) relaxation recently proposed by Rujeerapaiboon et al. (SIAM J Optim 29(2):1211-1239, 2019). However, this relaxation can be used in a branch-and-cut method only for small-size instances. Therefore, we derive a new SDP relaxation that scales better with the instance size and the number of clusters. In both cases, we strengthen the bound by adding polyhedral cuts. Benefiting from a tailored branching strategy which enforces pairwise constraints, we reduce the complexity of the problems arising in the children nodes. For the upper bound, instead, we present a local search procedure that exploits the solution of the SDP relaxation solved at each node. Computational results show that the proposed algorithm globally solves, for the first time, real-world instances of size 10 times larger than those solved by state-of-the-art exact methods.
In this work, we present approaches to rigorously certify A- and A(a)-stability ( a )-stability in Runge-Kutta methods through the solution of convex feasibility problems defined by linear matrix inequalities. We adop...
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In this work, we present approaches to rigorously certify A- and A(a)-stability ( a )-stability in Runge-Kutta methods through the solution of convex feasibility problems defined by linear matrix inequalities. We adopt two approaches. The first is based on sum-of-squares programming applied to the RungeKutta E-polynomial and is applicable to both A- and A(a)-stability. ( a )-stability. In the second, we sharpen the algebraic conditions for A-stability of Cooper, Scherer, T & uuml;rke, and Wendler to incorporate the Runge-Kutta order conditions. We demonstrate how the theoretical improvement enables the practical use of these conditions for certification of A-stability within a computational framework. We then use both approaches to obtain rigorous certificates of stability for several diagonally implicit schemes devised in the literature.
This paper proposes to integrate reconfigurable intelligent surfaces (RISs) with unmanned aerial vehicles (UAVs) as a resilience mechanism to mitigate outages in UAV networks due to UAV and link failures. The inherent...
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ISBN:
(数字)9798350368741
ISBN:
(纸本)9798350368758
This paper proposes to integrate reconfigurable intelligent surfaces (RISs) with unmanned aerial vehicles (UAVs) as a resilience mechanism to mitigate outages in UAV networks due to UAV and link failures. The inherent addition of RIS-aided links (UE-RIS-UAV links), combined with their reconfigurability, creates alternative paths for user equipment (UEs) to transmit signals to UAVs. The paper studies the problem of maximizing connectivity of UAV networks by jointly considering UE positioning, RIS-aided link selection, and phase shift design of RISs. To tackle it, we propose an efficient two-step solution. In the first step, we propose a supergradient method that locates the UEs in positions that improve their communication links until a certain connectivity threshold is satisfied. Given the optimized UE positioning, the second step jointly optimizes the RIS-aided link selection and RIS phase shift design using semidefinite programming (SDP). Through simulations, we illustrate the superiority of the proposed solution compared to the solutions available in the literature.
This article investigates robust optimality and duality for a class of nonsmooth semidefinite multiobjective programming problems with uncertain data (in short, UNSMP) via convexificators. Using the properties of conv...
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This article investigates robust optimality and duality for a class of nonsmooth semidefinite multiobjective programming problems with uncertain data (in short, UNSMP) via convexificators. Using the properties of convexificators, we deduce Fritz John (in short, FJ)-type and Karush-Kuhn-Tucker (in short, KKT)-type necessary optimality conditions for UNSMP. Moreover, under generalized convexity assumptions, we establish sufficient optimality criteria for UNSMP. Furthermore, we present the Wolfe-type (in short, WRD) and Mond-Weir-type (in short, MWRD) robust dual models corresponding to the primal problem UNSMP. Several illustrative non-trivial examples are furnished to demonstrate the significance of the established results.
We consider two min-max problems (1) minimizing the supremum of finitely many rational functions over a compact basic semi-algebraic set and (2) solving a 2-player zero-sum polynomial game in randomized strategies wit...
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We consider two min-max problems (1) minimizing the supremum of finitely many rational functions over a compact basic semi-algebraic set and (2) solving a 2-player zero-sum polynomial game in randomized strategies with compact basic semi-algebraic sets of pure strategies. In both problems the optimal value can be approximated by solving a hierarchy of semidefinite relaxations, in the spirit of the moment approach developed in Lasserre (SIAM J Optim 11:796-817, 2001;Math Program B 112:65-92, 2008). This provides a unified approach and a class of algorithms to compute Nash equilibria and min-max strategies of several static and dynamic games. Each semidefinite relaxation can be solved in time which is polynomial in its input size and practice on a sample of experiments reveals that few relaxations are needed for a good approximation (and sometimes even for finite convergence), a behavior similar to what was observed in polynomial optimization.
In this paper, we use fixed point theory and semidefinite programming to compute the performance bounds on convex block-sparsity recovery algorithms. As a prerequisite for optimal sensing matrix design, computable per...
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In this paper, we use fixed point theory and semidefinite programming to compute the performance bounds on convex block-sparsity recovery algorithms. As a prerequisite for optimal sensing matrix design, computable performance bounds would open doors for wide applications in sensor arrays, radar, DNA microarrays, and many other areas where block-sparsity arises naturally. We define a family of quality measures for arbitrary sensingmatrices as the optimal values of certain optimization problems. The reconstruction errors of convex recovery algorithms are bounded in terms of these quality measures. We demonstrate that as long as the number of measurements is relatively large, these quality measures are bounded away from zero for a large class of random sensing matrices, a result parallel to the probabilistic analysis of the block restricted isometry property. As the primary contribution of this work, we associate the quality measures with the fixed points of functions defined by a series of semidefinite programs. This relation with fixed point theory yields polynomia-ltime algorithms with global convergence guarantees to compute the quality measures.
We address the exact semidefinite programming feasibility problem (SDFP) consisting in checking that intersection of the cone of positive semidefinite matrices and some affine subspace of matrices with rational entrie...
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We address the exact semidefinite programming feasibility problem (SDFP) consisting in checking that intersection of the cone of positive semidefinite matrices and some affine subspace of matrices with rational entries is not empty. SDFP is a convex programming problem and is often considered as tractable since some of its approximate versions can be efficiently solved, e.g. by the ellipsoid algorithm. We prove that SDFP can decide comparison of numbers represented by the arithmetic circuits, i.e. circuits that use standard arithmetical operations as gates. Our reduction may give evidence to the intrinsic difficulty of SDFP (contrary to the common expectations) and clarify the complexity status of the exact SDP - an old open problem in the field of mathematical programming. (C) 2007 Elsevier B.V. All rights reserved.
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