Moment-SOS relaxation of the medium term hydrothermal dispatch problem

This issue's SIGEST paper, from the SIAM Journal on Optimization (SIOPT), is an excellent example of deep mathematical analysis, involving a range of mathematical techniques, which is having significant practical algorithmic and computational consequences. The title of the paper, “A Sum of Squares Approximation of Nonnegative Polynomials,” by Jean B. Lasserre, combines two concepts that are known to all SIAM readers: sums of squares, the types of functions that arise in many data fitting and other applications; and nonnegative polynomials, polynomials whose values are greater than or equal to zero for all inputs. It is obvious that any polynomial that can be written as a sum of squares is nonnegative. It is far from obvious whether any nonnegative polynomial has a representation as, or arbitrarily close to, a sum of squares of polynomials. In this SIGEST paper, Lasserre shows that this is the case. The nomination from the editors of SIOPT captures the immediate and more general contributions of this paper eloquently, in the context of a series of three papers of which this is the second: “Jean Lasserre, in a series of papers (starting from the first paper in SIOPT Vol. 11, pp. 796–817, and concluding with the third paper in SIOPT Vol. 15, pp. 383–393) by exploiting deep results from real algebraic geometry and theory of moments for multivariate polynomials, showed that the optimal value of a nonconvex polynomial optimization problem (i.e., an optimization problem whose objective function and constraints are specified by multivariate polynomials) can asymptotically be approximated by an increasing sequence of optimal values from increasing tighter semidefinite programming relaxation problems. Even though the result is asymptotic in the most general case, fortuitously, practical experience has revealed that very often, the first few levels of semidefinite programming relaxations typically can achieve the global optimum value. The theoretical connection that Jean Lasserre (and Pablo Parrilo at about the same time from the viewpoint of sum of squares representations of multivariate polynomials) had established between polynomial optimizations with real algebraic geometry and theory of moments has led to exciting developments in combinatorial optimization. In particular, many hard combinatorial problems such as the maximum clique problem of a graph and the quadratic assignment problem are now actively revisited by researchers under the framework of polynomial optimization to derive increasingly tighter bounds.” That is, a consequence of this and related work is the ability to use a sequence of semidefinite programming problems to (nearly) solve much more difficult optimization problems. Semidefinite programming problems are problems with a linear objective function and linear constraints in a matrix variable X, plus the additional constraint that X be positive semidefinite. Since a rich array of efficient algorithms for solving semidefinite programming problems has been developed in recent years, reducing more general problems to a sequence of semidefinite programming problems is an important advance. We hope that readers of SIAM Review will enjoy this glimpse into the sophisticated mathematics behind an important algorithmic innovation in optimization.

This work studies the problem of analyzing and, subsequently, optimizing the stabilization capabilities of a class of controllers for input constrained nonlinear systems. The proposed techniques apply to continuous, state feedback controllers which are defined in a subset of the state space where the time derivative of a known, candidate Control Lyapunov Function (CLF) can be made negative definite under the input constraints. This set is associated with the domain of this class of controllers and depends on the CLF. The analysis problem concerns approximating the domain and is posed via appropriately formulated set containment relationships through the generalized S-procedure with sum of squares (SOS) constraints. The optimization problem is concerned with the adjustment or enlargement of the domain and constitutes a way of controller synthesis. These objectives are pursued via optimizing over the coefficients of polynomial CLFs, through a sequence of semidefinite programming (SDP) problems with SOS constraints. By partitioning the state space based on the structure of the input value set and building upon earlier results on SOS methods, the SDP problems are subject to only convex constraints, rendering thus the proposed techniques computationally viable. The capabilities of the proposed algorithms are demonstrated through numerical examples.

In this paper, we consider a least square semidefinite programming problem under ellipsoidal data uncertainty. We show that the robustification of this uncertain problem can be reformulated as a semidefinite linear programming problem with an additional second-order cone constraint. We then provide an explicit quantitative sensitivity analysis on how the solution under the robustification depends on the size/shape of the ellipsoidal data uncertainty set. Next, we prove that, under suitable constraint qualifications, the reformulation has zero duality gap with its dual problem, even when the primal problem itself is infeasible. The dual problem is equivalent to minimizing a smooth objective function over the Cartesian product of second-order cones and the Euclidean space, which is easy to project onto. Thus, we propose a simple variant of the spectral projected gradient method (Birgin et al. in SIAM J. Optim. 10:1196---1211, 2000) to solve the dual problem. While it is well-known that any accumulation point of the sequence generated from the algorithm is a dual optimal solution, we show in addition that the dual objective value along the sequence generated converges to a finite value if and only if the primal problem is feasible, again under suitable constraint qualifications. This latter fact leads to a simple certificate for primal infeasibility in situations when the primal feasible set lies in a known compact set. As an application, we consider robust correlation stress testing where data uncertainty arises due to untimely recording of portfolio holdings. In our computational experiments on this particular application, our algorithm performs reasonably well on medium-sized problems for real data when finding the optimal solution (if exists) or identifying primal infeasibility, and usually outperforms the standard interior-point solver SDPT3 in terms of CPU time.

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.

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.

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.

This paper deals with the problem of estimating the steering direction of a signal, embedded in Gaussian disturbance, under a general quadratic inequality constraint, representing the uncertainty region of the steering. We resort to the maximum likelihood (ML) criterion and focus on two scenarios. The former assumes that the complex amplitude of the useful signal component fluctuates from snapshot to snapshot. The latter supposes that the useful signal keeps a constant amplitude within all the snapshots. We prove that the ML criterion leads in both cases to a fractional quadratically constrained quadratic problem (QCQP). In order to solve it, we first relax the problem into a constrained fractional semidefinite programming (SDP) problem which is shown equivalent, via the Charnes-Cooper transformation, to an SDP problem. Then, exploiting a suitable rank-one decomposition, we show that the SDP relaxation is tight and give a procedure to construct (in polynomial time) an optimal solution of the original problem from an optimal solution of the fractional SDP. We also assess the quality of the derived estimator through a comparison between its performance and the constrained Cramer Rao lower Bound (CRB). Finally, we give two applications of the proposed theoretical framework in the context of radar detection.

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

Semidefinite programming (SDP) covers a wide range of applications such as robust optimization, polynomial optimization, combinatorial optimization, system and control theory, financial engineering, machine learning, quantum information and quantum chemistry. In those applications, SDP problems can be large scale easily. Such large scale SDP problems often satisfy a certain sparsity characterized by a chordal graph structure. This sparsity is classified in two types. The one is the domain space sparsity (d-space sparsity) for positive semidefinite symmetric matrix variables involved in SDP problems, and the other the range space sparsity (r-space sparsity) for matrix-inequality constraints in SDP problems. In this short note, we survey how we exploit these two types of sparsities to solve large scale linear and nonlinear SDP problems. We refer to the paper [7] for more details.

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.

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 sensor network localization (SNL) problem in embedding dimension r consists of locating the positions of wireless sensors, given only the distances between sensors that are within radio range and the positions of a subset of the sensors (called anchors). Current solution techniques relax this problem to a weighted, nearest, (positive) semidefinite programming (SDP) completion problem by using the linear mapping between Euclidean distance matrices (EDM) and semidefinite matrices. The resulting SDP is solved using primal-dual interior point solvers, yielding an expensive and inexact solution. This relaxation is highly degenerate in the sense that the feasible set is restricted to a low dimensional face of the SDP cone, implying that the Slater constraint qualification fails. Cliques in the graph of the SNL problem give rise to this degeneracy in the SDP relaxation. In this paper, we take advantage of the absence of the Slater constraint qualification and derive a technique for the SNL problem, with exact data, that explicitly solves the corresponding rank restricted SDP problem. No SDP solvers are used. For randomly generated instances, we are able to efficiently solve many huge instances of this NP-hard problem to high accuracy by finding a representation of the minimal face of the SDP cone that contains the SDP matrix representation of the EDM. The main work of our algorithm consists in repeatedly finding the intersection of subspaces that represent the faces of the SDP cone that correspond to cliques of the SNL problem.

Representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and semidefinite programming (SDP) relaxation of polynomial optimization problems. We discuss effective methods to obtain a simpler representation of a “sparse” polynomial as a sum of squares of sparse polynomials by eliminating redundancy.

Sums of squares based approximation algorithms for MAX-SAT

We consider a dynamic time division duplexing wireless network and propose a distributed coordinated beamforming algorithm based on Alternating Direction Method of Multipliers (ADMM) technique assuming the availability of perfect channel state information. Our design objective is to minimize the sum transmit power at the base stations subject to minimum signal-to-interference-plus-noise ratio (SINR) constraints for downlink mobile stations and a maximum interference power threshold for uplink mobile stations. First, we propose a centralized algorithm based on the relaxed Semidefinite Programming (SDP) technique. To obtain the beamforming solution in a distributed way, we further propose a distributed coordinated beamforming algorithm using the ADMM technique. Detailed simulation results are presented to examine the effectiveness of the proposed algorithms. It is shown that the proposed algorithm achieves better performance in terms of the design objective and converges faster than the reference algorithm based on primal decomposition.