Facet-defining triangle inequalities for the quadratic linear ordering problem

A computational study for bilevel quadratic programs using semidefinite relaxations

A sensor network localization problem can be formulated as a quadratic optimization problem (QOP). For QOPs, semidefinite programming (SDP) relaxation by Lasserre with relaxation order 1 for general polynomial optimization problems (POPs) is known to be equivalent to the sparse SDP relaxation by Waki et al. with relaxation order 1, except for the size and sparsity of the resulting SDP relaxation problems. We show that the sparse SDP relaxation applied to the QOP is at least as strong as the Biswas–Ye SDP relaxation for the sensor network localization problem. A sparse variant of the Biswas–Ye SDP relaxation, which is equivalent to the original Biswas–Ye SDP relaxation, is also derived. We compare numerically the sparse SDP relaxation applied to the QOP, the Biswas–Ye SDP relaxation, its sparse variant, and the edge-based SDP relaxation by Wang et al. to confirm the effectiveness of the proposed techniques for exploiting the sparsity in SDP relaxation for sensor network localization problems. The sparse SDP relaxation applied to the QOP is much faster than the Biswas–Ye SDP relaxation, and the sparse variant of the Biswas–Ye SDP relaxation outperforms all other SDP relaxations in speed.

In this paper, we consider binary quadratically constrained quadratic problems and propose a new approach to generate stronger bounds than the ones obtained using the Semidefinite Programming relaxation. The new relaxation is based on the Boolean Quadric Polytope and is solved via a Dantzig–Wolfe Reformulation in matrix space. For block-decomposable problems, we extend the relaxation and analyze the theoretical properties of this novel approach. If overlapping size of blocks is at most two (i.e., when the sparsity graph of any pair of intersecting blocks contains either a cut node or an induced diamond graph), we establish equivalence to the one based on the Boolean Quadric Polytope. We prove that this equivalence does not hold if the sparsity graph is not chordal and we conjecture that equivalence holds for any block structure with a chordal sparsity graph. The tailored decomposition algorithm in the matrix space is used for efficiently bounding sparsely structured problems. Preliminary numerical results show that the proposed approach yields very good bounds in reasonable time.

Statistical inference problems arising within signal processing, data mining, and machine learning naturally give rise to hard combinatorial optimization problems. These problems become intractable when the dimensionality of the data is large, as is often the case for modern datasets. A popular idea is to construct convex relaxations of these combinatorial problems, which can be solved efficiently for large-scale datasets. Semidefinite programming (SDP) relaxations are among the most powerful methods in this family and are surprisingly well suited for a broad range of problems where data take the form of matrices or graphs. It has been observed several times that when the statistical noise is small enough, SDP relaxations correctly detect the underlying combinatorial structures. In this paper we develop asymptotic predictions for several detection thresholds, as well as for the estimation error above these thresholds. We study some classical SDP relaxations for statistical problems motivated by graph synchronization and community detection in networks. We map these optimization problems to statistical mechanics models with vector spins and use nonrigorous techniques from statistical mechanics to characterize the corresponding phase transitions. Our results clarify the effectiveness of SDP relaxations in solving high-dimensional statistical problems.

This paper is concerned with the analysis and comparison of semidefinite programming (SDP) relaxations for the satisfiability (SAT) problem. Our presentation is focussed on the special case of 3-SAT, but the ideas presented can in principle be extended to any instance of SAT specified by a set of boolean variables and a propositional formula in conjunctive normal form. We propose a new SDP relaxation for 3-SAT and prove some of its theoretical properties. We also show that, together with two SDP relaxations previously proposed in the literature, the new relaxation completes a trio of linearly sized relaxations with increasing rank-based guarantees for proving satisfiability. A comparison of the relative practical performances of the SDP relaxations shows that, among these three relaxations, the new relaxation provides the best tradeoff between theoretical strength and practical performance within an enumerative algorithm.

The optimal power flow (OPF) problem determines an optimal operating point of the power network that minimizes a certain objective function subject to physical and network constraints. There has been a great deal of attention in convex relaxation of the OPF problem in recent years, and it has been shown that a semidefinite programming (SDP) relaxation can solve different classes of the nonconvex OPF problem to global or near-global optimality. Although it is known that the SDP relaxation is exact for radial networks under various conditions, solving the OPF problem for cyclic networks needs further research. In this paper, we propose sufficient conditions under which the SDP relaxation is exact for special but important cyclic networks. More precisely, when the objective function is a linear function of active powers, we show that the SDP relaxation is exact for odd cycles under certain conditions. Also, the exactness of the SDP relaxation for simple cycles of size 3 and 4 is proved under different technical conditions. The existence of rank-1 or -2 SDP solutions for weakly-cyclic networks is also proved in both lossy and lossless cases. In addition, when the objective function is an increasing function of reactive powers, we prove that the SDP relaxation is exact under certain conditions. This result justifies why the sum of reactive powers acts as a low-rank-promoting term for OPF. The findings of this paper provide intuition into the behavior of SDPs for different building blocks of cyclic networks.

A polyhedral approach for a constrained quadratic 0–1 problem

It has been recently proven that the semidefinite programming (SDP)\nrelaxation of the optimal power flow problem over radial networks is exact\nunder technical conditions such as not including generation lower bounds or\nallowing load over-satisfaction. In this paper, we investigate the situation\nwhere generation lower bounds are present. We show that even for a two-bus\none-generator system, the SDP relaxation can have all possible approximation\noutcomes, that is (1) SDP relaxation may be exact or (2) SDP relaxation may be\ninexact or (3) SDP relaxation may be feasible while the OPF instance may be\ninfeasible. We provide a complete characterization of when these three\napproximation outcomes occur and an analytical expression of the resulting\noptimality gap for this two-bus system. In order to facilitate further\nresearch, we design a library of instances over radial networks in which the\nSDP relaxation has positive optimality gap. Finally, we propose valid\ninequalities and variable bound tightening techniques that significantly\nimprove the computational performance of a global optimization solver. Our work\ndemonstrates the need of developing efficient global optimization methods for\nthe solution of OPF even in the simple but fundamental case of radial networks.\n

Based on the special kind of spreading sequence and semidefinite programming relaxation, A new multiuser detector for asynchronous code division multiple access system is presented. At each transmitter, the special spreading sequence whose second half is the replica of the first is employed. Maximum-likelihood(ML) multiuser detection only need detect 2K once. Then ML detection can be efficiently and accurately approximated using the semidefinite relaxation. Its time complexity per bit decrease to the order of O(2K <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3.5</sup> ) , while the Viterbi Algorithm (optimum detector) requires a exponentially computational complexity. We compare the performances of the new SDP method and a joint method of block coordinate ascent and semidefinite programming relaxation(BCD-SDR) method for the asynchronous multiuser detection problem in various situations, and simulations demonstrate that the new semidefinite programming method often yields better bit error rate (BER) performances than the BCD-SDR method, but the average CPU time of this method is significantly reduced.

We investigate the notion of stability proposed by Bilu and Linial. We obtain an exact polynomial-time algorithm for γ-stable Max Cut instances with for some absolute constant c > 0. Our algorithm is robust: it never returns an incorrect answer; if the instance is γ-stable, it finds the maximum cut, otherwise, it either finds the maximum cut or certifies that the instance is not γ-stable. We prove that there is no robust polynomial-time algorithm for γ-stable instances of Max Cut when γ < αSC(n/2), where αSC is the best approximation factor for Sparsest Cut with non-uniform demands. That suggests that solving γ-stable instances with might be difficult or even impossible. Our algorithm is based on semidefinite programming. We show that the standard SDP relaxation for Max Cut (with triangle inequalities) is integral if , where is the least distortion with which every n point metric space of negative type embeds into ℓ1. On the negative side, we show that the SDP relaxation is not integral when . Moreover, there is no tractable convex relaxation for γ-stable instances of Max Cut when γ < αSC(n/2). Our results significantly improve previously known results. The best previously known algorithm for γ-stable instances of Max Cut required that (for some c > 0) [Bilu, Daniely, Linial, and Saks]. No hardness results were known for the problem. Additionally, we present an exact robust polynomial-time algorithm for 4-stable instances of Minimum Multiway Cut. We also study a relaxed notion of weak stability and present algorithms for weakly stable instances of Max Cut and Minimum Multiway Cut.

The SDP (semidefinite programming) relaxation for general POPs (polynomial optimization problems), which was proposed as a method for computing global optimal solutions of POPs by Lasserre, has become an active research subject recently. We propose a new heuristic method exploiting the equality constraints in a given POP, and strengthen the SDP relaxation so as to achieve faster convergence to the global optimum of the POP. We can apply this method to both of the dense SDP relaxation which was originally proposed by Lasserre, and the sparse SDP relaxation which was later proposed by Kim, Kojima, Muramatsu and Waki. Especially, our heuristic method incorporated into the sparse SDP relaxation method has shown a promising performance in numerical experiments on large scale sparse POPs. Roughly speaking, we induce valid equality constraints from the original equality constraints of the POP, and then use them to convert the dense or sparse SDP relaxation into a new stronger SDP relaxation. Our method is enlightened by some strong theoretical results on the convergence of SDP relaxations for POPs with equality constraints provided by Lasserre , Parrilo and Laurent, but we place the main emphasis on the practical aspect to compute more accurate lower bounds of larger sparse POPs.

A Semidefinite Programming (SDP) relaxation is an effective computational method to solve a Sensor Network Localization problem, which attempts to determine the locations of a group of sensors given the distances between some of them. In this paper, we analyze and determine new sufficient conditions and formulations that guarantee that the SDP relaxation is exact, i.e., gives the correct solution. These conditions can be useful for designing sensor networks and managing connectivities in practice. Our main contribution is threefold: First, we present the first non-asymptotic bound on the connectivity (or radio) range requirement of randomly distributed sensors in order to ensure the network is uniquely localizable with high probability. Determining this range is a key component in the design of sensor networks, and we provide a result that leads to a correct localization of each sensor, for any number of sensors. Second, we introduce a new class of graphs that can always be correctly localized by an SDP relaxation. Specifically, we show that adding a simple objective function to the SDP relaxation model will ensure that the solution is correct when applied to a triangulation graph. Since triangulation graphs are very sparse, this is informationally efficient, requiring an almost minimal amount of distance information. Finally, we analyze a number of objective functions for the SDP relaxation to solve the localization problem for a general graph.Key wordsSensor network localizationGraph realizationSemidefinite programmingSubject Classifications90C2290C4690C9090B18

In this paper, we propose a new robust analysis tool motivated by large-scale systems. The ℋ∞ norm of a system measures its robustness by quantifying the worst-case behavior of a system perturbed by a unit-energy disturbance. However, the disturbance that induces such worst-case behavior requires perfect coordination among all disturbance channels. Given that many systems of interest, such as the power grid, the internet and automated vehicle platoons, are large-scale and spatially distributed, such coordination may not be possible, and hence the ℋ∞ norm, used as a measure of robustness, may be too conservative. We therefore propose a cardinality constrained variant of the ℋ∞ norm in which an adversarial disturbance can use only a limited number of channels. As this problem is inherently combinatorial, we present a semidefinite programming (SDP) relaxation based on the l1 norm that yields an upper bound on the cardinality constrained robustness problem. We further propose a simple rounding heuristic based on the optimal solution of our SDP relaxation, which provides a corresponding lower bound. Motivated by privacy in large-scale systems, we also extend these relaxations to computing the minimum gain of a system subject to a limited number of inputs. Finally, we also present a SDP based optimal controller synthesis method for minimizing the SDP relaxation of our novel robustness measure. The effectiveness of our semidefinite relaxation is demonstrated through numerical examples.

In this paper we study a class of quadratic maximization problems and their semidefinite programming (SDP) relaxation. For a special subclass of the problems we show that the SDP relaxation provides an exact optimal solution. Another subclass, which is ??-hard, guarantees that the SDP relaxation yields an approximate solution with a worst-case performance ratio of 0.87856.... This is a generalization of the well-known result of Goemans and Williamson for the maximum-cut problem. Finally, we discuss extensions of these results in the presence of a certain type of sign restrictions.

A method for quickly bounding the optimal objective value of an OPF problem using a semidefinite relaxation and a local solution