A Multigrid Approach to SDP Relaxations of Sparse Polynomial Optimization Problems

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

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.

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.

Cooperative beamforming in relay networks is considered, in which a source transmits to its destination with the help of a set of cooperating nodes. The source first transmits locally. The cooperating nodes that receive the source signal retransmit a weighted version of it in an amplify-and-forward (AF) fashion. Assuming knowledge of the second-order statistics of the channel state information, beamforming weights are determined so that the signal-to-noise ratio (SNR) at the destination is maximized subject to two different power constraints, i.e., a total (source and relay) power constraint, and individual relay power constraints. For the former constraint, the original problem is transformed into a problem of one variable, which can be solved via Newton's method. For the latter constraint, the original problem is transformed into a homogeneous quadratically constrained quadratic programming (QCQP) problem. In this case, it is shown that when the number of relays does not exceed three the global solution can always be constructed via semidefinite programming (SDP) relaxation and the matrix rank-one decomposition technique. For the cases in which the SDP relaxation does not generate a rank one solution, two methods are proposed to solve the problem: the first one is based on the coordinate descent method, and the second one transforms the QCQP problem into an infinity norm maximization problem in which a smooth finite norm approximation can lead to the solution using the augmented Lagrangian method.

A polynomial optimization problem (POP) consists of minimizing a multivariate real polynomial on a semi-algebraic set \(K\) described by polynomial inequalities and equations. In its full generality it is a non-convex, multi-extremal, difficult global optimization problem. More than an decade ago, J. B. Lasserre proposed to solve POPs by a hierarchy of convex semidefinite programming (SDP) relaxations of increasing size. Each problem in the hierarchy has a primal SDP formulation (a relaxation of a moment problem) and a dual SDP formulation (a sum-of-squares representation of a polynomial Lagrangian of the POP). In this note, we show that there is no duality gap between each primal and dual SDP problem in Lasserre’s hierarchy, provided one of the constraints in the description of set \(K\) is a ball constraint. Our proof uses elementary results on SDP duality, and it does not assume that \(K\) has a strictly feasible point.

Identification Aspects of SDP Based Polynomial Optimization Relaxations

We consider the problem of estimating the discrete clustering structures under the sub-Gaussian mixture model. Our main results establish a hidden integrality property of a semidefinite programming (SDP) relaxation for this problem: while the optimal solution to the SDP is not integer-valued in general, its estimation error can be upper bounded by that of an idealized integer program. The error of the integer program, and hence that of the SDP, are further shown to decay exponentially in the signal-to-noise ratio. In addition, we show that the SDP relaxation is robust under the semirandom setting in which an adversary can modify the data generated from the mixture model. In particular, we generalize the hidden integrality property to the semirandom model and thereby show that SDP achieves the optimal error bound in this setting. These results together highlight the “global-to-local” mechanism that drives the performance of the SDP relaxation. To the best of our knowledge, our result is the first exponentially decaying error bound for convex relaxations of mixture models. A corollary of our results shows that in certain regimes, the SDP solutions are in fact integral and exact. More generally, our results establish sufficient conditions for the SDP to correctly recover the cluster memberships of [Formula: see text] fraction of the points for any [Formula: see text]. As a special case, we show that under the [Formula: see text]-dimensional stochastic ball model, SDP achieves nontrivial (sometimes exact) recovery when the center separation is as small as [Formula: see text], which improves upon previous exact recovery results that require constant separation.

We propose a hierarchy of semidefinite programming (SDP) relaxations for polynomial optimization with sparse patterns over unbounded feasible sets. The convergence of the proposed SDP hierarchy is established for a class of polynomial optimization problems. This is done by employing known sums-of-squares sparsity techniques of Kojima and Muramatsu Comput Optim Appl 42(1):31---41, (2009) and Lasserre SIAM J Optim 17:822---843, (2006) together with a representation theorem for polynomials over unbounded sets obtained recently in Jeyakumar et al. J Optim Theory Appl 163(3):707---718, (2014). We demonstrate that the proposed sparse SDP hierarchy can solve some classes of large scale polynomial optimization problems with unbounded feasible sets using the polynomial optimization solver SparsePOP developed by Waki et al. ACM Trans Math Softw 35:15 (2008).

A polynomial optimization problem whose objective function is represented as a sum of positive and even powers of polynomials, called a polynomial least squares problem, is considered. Methods to transform a polynomial least square problem to polynomial semidefinite programs to reduce degrees of the polynomials are discussed. Computational efficiency of solving the original polynomial least squares problem and the transformed polynomial semidefinite programs is compared. Numerical results on selected polynomial least square problems show better computational performance of a transformed polynomial semidefinite program, especially when degrees of the polynomials are larger.

We consider a property of positive polynomials on a compact set with a small perturbation. When applied to a polynomial optimization problem (POP), the property implies that the optimal value of the corresponding SemiDefinite Programming (SDP) relaxation with sufficiently large relaxation order is bounded from below by and from above by , where is the optimal value of the POP. We propose new SDP relaxations for POP based on modifications of existing sums-of-squares representation theorems. An advantage of our SDP relaxations is that in many cases they are of considerably smaller dimension than those originally proposed by Lasserre. We present some applications and the results of our computational experiments.

Markov Random Fields (MRFs) are a popular model for several pattern recognition and reconstruction problems in robotics and computer vision. Inference in MRFs is intractable in general and related work resorts to approximation algorithms. Among those techniques, semidefinite programming (SDP) relaxations have been shown to provide accurate estimates while scaling poorly with the problem size and being typically slow for practical applications. Our first contribution is to design a dual ascent method to solve standard SDP relaxations that takes advantage of the geometric structure of the problem to speed up computation. This technique, named Dual Ascent Riemannian Staircase (DARS), is able to solve large problem instances in seconds. Our second contribution is to develop a second and faster approach. The backbone of this second approach is a novel SDP relaxation combined with a fast and scalable solver based on smooth Riemannian optimization. We show that this approach, named Fast Unconstrained SEmidefinite Solver (FUSES), can solve large problems in milliseconds. Contrarily to local MRF solvers, e.g., loopy belief propagation, our approaches do not require an initial guess. Moreover, we leverage recent results from optimization theory to provide per-instance sub-optimality guarantees. We demonstrate the proposed approaches in multi-class image segmentation problems. Extensive experimental evidence shows that (i) FUSES and DARS produce near-optimal solutions, attaining an objective within 0.1% of the optimum, (ii) FUSES and DARS are remarkably faster than general-purpose SDP solvers, and FUSES is more than two orders of magnitude faster than DARS while attaining similar solution quality, (iii) FUSES is faster than local search methods while being a global solver.

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

MISO NOMA downlink beamforming optimization with per-antenna power constraints

This paper studies the so-called biquadratic optimization over unit spheres $\min_{x\in \mathbb{R}^n,y\in \mathbb{R}^m}\sum_{1\leq i,k\leq n,\,1\leq j,l \leq m}b_{ijkl}x_{i}y_{j}x_{k}{y}_l$, subject to $\|x\| = 1$, $\|y\| =1$. We show that this problem is NP-hard, and there is no polynomial time algorithm returning a positive relative approximation bound. Then, we present various approximation methods based on semidefinite programming (SDP) relaxations. Our theoretical results are as follows: For general biquadratic forms, we develop a $\frac{1}{2\max\{m,n\}^2}$-approximation algorithm under a slightly weaker approximation notion; for biquadratic forms that are square-free, we give a relative approximation bound $\frac{1}{nm}$; when $\min\{n,m\}$ is a constant, we present two polynomial time approximation schemes (PTASs) which are based on sum of squares (SOS) relaxation hierarchy and grid sampling of the standard simplex. For practical computational purposes, we propose the first order SOS relaxation, a convex quadratic SDP relaxation, and a simple minimum eigenvalue method and show their error bounds. Some illustrative numerical examples are also included.

Optimal power flow schedules with reduced low-frequency oscillations