We consider semidefinite programming (SDP) approaches for solving the maximum satisfiability (MAX-SAT) problem and weighted partial MAX-SAT. It is widely known that SDP is well-suited to approximate (MAX-)2-SAT. Our w...
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We consider semidefinite programming (SDP) approaches for solving the maximum satisfiability (MAX-SAT) problem and weighted partial MAX-SAT. It is widely known that SDP is well-suited to approximate (MAX-)2-SAT. Our work shows the potential of SDP also for other satisfiability problems by being competitive with some of the best solvers in the yearly MAX-SAT competition. Our solver combines sum of squares (SOS)-based SDP bounds and an efficient parser within a branch-and-bound scheme. On the theoretical side, we propose a family of semidefinite feasibility problems and show that a member of this family provides the rank-two guarantee. We also provide a parametric family of semidefinite relaxations for MAX-SAT and derive several properties of monomial bases used in the SOS approach. We connect two well-known SDP approaches for (MAX)-SAT in an elegant way. Moreover, we relate our SOS-SDP relaxations for partial MAX-SAT to the known SAT relaxations.
We study the problem of distinguishing quantum states using local operations and classical communication (LOCC). A question of fundamental interest is whether there exist sets of k <= d orthogonal maximally entangl...
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We study the problem of distinguishing quantum states using local operations and classical communication (LOCC). A question of fundamental interest is whether there exist sets of k <= d orthogonal maximally entangled states in C-d circle times C-d that are not perfectly distinguishable by LOCC. A recent result by Yu, Duan, and Ying [Phys. Rev. Lett. 109 020506 (2012)] gives an affirmative answer for the case k = d. We give, for the first time, a proof that such sets of states indeed exist even in the case k < d. Our result is constructive and holds for an even wider class of operations known as positive-partial-transpose measurements (PPT). The proof uses the characterization of the PPT-distinguishability problem as a semidefinite program.
Target localization plays an indispensable role in wireless network and array signal processing. Generally, the traditional methods are restricted to the line of sight (LOS) transmission link assumption, and may fail ...
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Target localization plays an indispensable role in wireless network and array signal processing. Generally, the traditional methods are restricted to the line of sight (LOS) transmission link assumption, and may fail in the non line of sight (NLOS) scenario. As a promising technology, reconfigurable intelligent surface (RIS) can customize the wireless channel and then overcome the performance degradation due to the lack of LOS. In this paper, we consider the power optimization problem of RIS localization, which is crucial for localization accuracy and energy consumption. In order to minimize the transmit power, we investigate several power optimization problems of the RIS localization under the constraints of the localization accuracy and the phase shift parameters of the RIS. Specifically, aiming at the RIS system with an anchor node (AN) and an target node (TN), we derive the Cramer-Rao bound (CRB) in terms of the location parameters of the TN, and prove the theoretic result of the relation between the optimal power and the phase parameters of the RIS. Based on these results, we formulate the corresponding power optimization problem, and derive the analytical solution of the optimization problem. Next, we obtain the convex form of the power optimization problem for the second scenario, that is, the RIS system with multiple TNs and single AN. Lastly, for the multiple ANs scenario with single TN, we solve the power optimization problem by using semidefinite release (SDR) method. Simulation results verify the feasibility of the proposed methods. (c) 2021 Elsevier B.V. All rights reserved.
Quantum query complexity is pivotal in the analysis of quantum algorithms, encompassing well-known examples like search and period-finding algorithms. These algorithms typically involve a sequence of unitary operation...
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Quantum query complexity is pivotal in the analysis of quantum algorithms, encompassing well-known examples like search and period-finding algorithms. These algorithms typically involve a sequence of unitary operations and oracle calls dependent on an input variable. In this study, we introduce a variational learning approach to explore quantum query complexity. Our method employs an efficient parameterization of the unitary operations and utilizes a loss function derived from the algorithm's error probability. We apply this technique to various quantum query complexities, notably devising a new algorithm that resolves the 5-bit Hamming modulo problem with four queries, addressing an open question from Cornelissen et al (2021 arXiv:2112.14682). This finding is corroborated by a semidefinite programming (SDP) approach. Our numerical method exhibits superior memory efficiency compared to SDP and can identify quantum query algorithms (QQAs) that require a smaller workspace register dimension, an aspect not optimized by SDP. These advancements present a significant step forward in the practical application and understanding of QQAs.
Chordal decomposition techniques are used to reduce large structured positive semidefinite matrix constraints in semidefinite programs (SDPs). The resulting equivalent problem contains multiple smaller constraints on ...
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Chordal decomposition techniques are used to reduce large structured positive semidefinite matrix constraints in semidefinite programs (SDPs). The resulting equivalent problem contains multiple smaller constraints on the nonzero blocks (or cliques) of the original problem matrices. This usually leads to a significant reduction in the overall solve time. A further reduction is possible by remerging cliques with significant overlap. The degree of overlap for which this is effective is dependent on the particular solution algorithm and hardware to be employed. We propose a novel clique merging approach that utilizes the clique graph to identify suitable merge candidates and that is suitable for any SDP solver algorithm. We show its performance in combination with a first-order method by comparing it with two existing approaches on selected problems from a benchmark library. Our approach is implemented in the latest version of the conic ADMM-solver COSMO.
We study the structure of the set of algebraic curvature operators satisfying a sectional curvature bound under the light of the emerging field of convex algebraic geometry. More precisely, we determine in which dimen...
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We study the structure of the set of algebraic curvature operators satisfying a sectional curvature bound under the light of the emerging field of convex algebraic geometry. More precisely, we determine in which dimensions n this convex semialgebraic set is a spectrahedron or a spectrahedral shadow;in particular, for n >= 5, these give new counterexamples to the Helton-Nie conjecture. Moreover, efficient algorithms are provided if n = 4 to test membership in such a set. For n >= 5, algorithms using semidefinite programming are obtained from hierarchies of inner approximations by spectrahedral shadows and outer relaxations by spectrahedra.
One of the current challenges of risk modelling consists in building global risk models from local ones: from a set of local market risk forecasts (local covariance matrices) and cross-market correlations, a global co...
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One of the current challenges of risk modelling consists in building global risk models from local ones: from a set of local market risk forecasts (local covariance matrices) and cross-market correlations, a global covariance matrix preserving local market estimations and restoring a positive semidefinite matrix must be computed. Convex optimisation, taking advantage of the convex properties of dual functions, is an original and high-performing approach for such a process. In this paper, a particular semidefinite program is posed and solved with dual convex algorithms for correlation matrices in order to build a global risk model, starting from a set local market covariance, and cross-correlation. Some numerical illustrations are given.
A wide variety of problems involving analysis of systems can be rewritten as a semidefinite program. When solving these problems optimization algorithms are used. Large size makes the problems unsolvable in practice a...
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A wide variety of problems involving analysis of systems can be rewritten as a semidefinite program. When solving these problems optimization algorithms are used. Large size makes the problems unsolvable in practice and computationally more effective solvers are needed. This paper investigates how to exploit structure and problem knowledge in some control applications. It is shown that inexact search directions are useful to reduce the computational burden and that operator formalism can be utilized to derive tailored calculations.
This paper proposes a new semidefinite programming relaxation for the satisfiability problem. This relaxation is an extension of previous relaxations arising from the paradigm of partial semidefinite liftings for 0/1 ...
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This paper proposes a new semidefinite programming relaxation for the satisfiability problem. This relaxation is an extension of previous relaxations arising from the paradigm of partial semidefinite liftings for 0/1 optimization problems. The construction of the relaxation depends on a choice of permutations of the clauses, and different choices may lead to different relaxations. We then consider the Tseitin instances, a class of instances known to be hard for certain proof systems, and prove that for any choice of permutations, the proposed relaxation is exact for these instances, meaning that a Tseitin instance is unsatisfiable if and only if the corresponding semidefinite programming relaxation is infeasible.
semidefinite optimization, commonly referred to as semidefinite programming, has been a remarkably active area of research in optimization during the last decade. For combinatorial problems in particular, semidefinite...
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semidefinite optimization, commonly referred to as semidefinite programming, has been a remarkably active area of research in optimization during the last decade. For combinatorial problems in particular, semidefinite programming has had a truly significant impact. This paper surveys some of the results obtained in the application of semidefinite programming to satisfiability and maximum-satisfiability problems. The approaches presented in some detail include the ground-breaking approximation algorithm of Goemans and Williamson for MAX-2-SAT, the Gap relaxation of de Klerk, van Maaren and Warners, and strengthenings of the Gap relaxation based on the Lasserre hierarchy of semidefinite liftings for polynomial optimization problems. We include theoretical and computational comparisons of the aforementioned semidefinite relaxations for the special case of 3-SAT, and conclude with a review of the most recent results in the application of semidefinite programming to SAT and MAX-SAT.
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