Analysis and synthesis of nonlinear controllers for input constrained systems using semidefinite programming optimization

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.

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.

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.

Moment-SOS relaxation of the medium term hydrothermal dispatch problem

We revisit the problem of protein structure determination from geometrical restraints from NMR, using convex optimization. It is well-known that the NP-hard distance geometry problem of determining atomic positions from pairwise distance restraints can be relaxed into a convex semidefinite program (SDP). However, often the NOE distance restraints are too imprecise and sparse for accurate structure determination. Residual dipolar coupling (RDC) measurements provide additional geometric information on the angles between atom-pair directions and axes of the principal-axis-frame. The optimization problem involving RDC is highly non-convex and requires a good initialization even within the simulated annealing framework. In this paper, we model the protein backbone as an articulated structure composed of rigid units. Determining the rotation of each rigid unit gives the full protein structure. We propose solving the non-convex optimization problems using the sum-of-squares (SOS) hierarchy, a hierarchy of convex relaxations with increasing complexity and approximation power. Unlike classical global optimization approaches, SOS optimization returns a certificate of optimality if the global optimum is found. Based on the SOS method, we proposed two algorithms-RDC-SOS and RDC-NOE-SOS, that have polynomial time complexity in the number of amino-acid residues and run efficiently on a standard desktop. In many instances, the proposed methods exactly recover the solution to the original non-convex optimization problem. To the best of our knowledge this is the first time SOS relaxation is introduced to solve non-convex optimization problems in structural biology. We further introduce a statistical tool, the Cramér-Rao bound (CRB), to provide an information theoretic bound on the highest resolution one can hope to achieve when determining protein structure from noisy measurements using any unbiased estimator. Our simulation results show that when the RDC measurements are corrupted by Gaussian noise of realistic variance, both SOS based algorithms attain the CRB. We successfully apply our method in a divide-and-conquer fashion to determine the structure of ubiquitin from experimental NOE and RDC measurements obtained in two alignment media, achieving more accurate and faster reconstructions compared to the current state of the art.

Solving optimization problems with sparse or low-rank optimal solutions has been an important topic since the recent emergence of compressed sensing and its matrix extensions such as the matrix rank minimization and robust principal component analysis problems. Compressed sensing enables one to recover a signal or image with fewer observations than the “length” of the signal or image, and thus provides potential breakthroughs in applications where data acquisition is costly. However, the potential impact of compressed sensing cannot be realized without efficient optimization algorithms that can handle extremely large-scale and dense data from real applications. Although the convex relaxations of these problems can be reformulated as either linear programming, second-order cone programming or semidefinite programming problems, the standard methods for solving these relaxations are not applicable because the problems are usually of huge size and contain dense data. In this dissertation, we give efficient algorithms for solving these “sparse” optimization problems and analyze the convergence and iteration complexity properties of these algorithms. Chapter 2 presents algorithms for solving the linearly constrained matrix rank minimization problem. The tightest convex relaxation of this problem is the linearly constrained nuclear norm minimization. Although the latter can be cast and solved as a semidefinite programming problem, such an approach is computationally expensive when the matrices are large. In Chapter 2, we propose fixed-point and Bregman iterative algorithms for solving the nuclear norm minimization problem and prove convergence of the first of these algorithms. By using a homotopy approach together with an approximate singular value decomposition procedure, we get a very fast, robust and powerful algorithm, which we call FPCA (Fixed Point Continuation with Approximate SVD), that can solve very large matrix rank minimization problems. Our numerical results on randomly generated and real matrix completion problems demonstrate that this algorithm is much faster and provides much better recoverability than semidefinite programming solvers such as SDPT3. For example, our algorithm can recover 1000 × 1000 matrices of rank 50 with a relative error of 10−5 in about 3 minutes by sampling only 20 percent of the elements. We know of no other method that achieves as good recoverability. Numerical experiments on online recommendation, DNA microarray data set and image inpainting problems demonstrate the effectiveness of our algorithms. In Chapter 3, we study the convergence/recoverability properties of the fixed-point continuation algorithm and its variants for matrix rank minimization. Heuristics for determining the rank of the matrix when its true rank is not known are also proposed. Some of these algorithms are closely related to greedy algorithms in compressed sensing. Numerical results for these algorithms for solving linearly constrained matrix rank minimization problems are reported. Chapters 4 and 5 considers alternating direction type methods for solving composite convex optimization problems. We present in Chapter 4 alternating linearization algorithms that are based on an alternating direction augmented Lagrangian approach for minimizing the sum of two convex functions. Our basic methods require at most O(1/e) iterations to obtain an e-optimal solution, while our accelerated (i.e., fast) versions require at most O(1/ 3 ) iterations, with little change in the computational effort required at each iteration. For more general problem, i.e., minimizing the sum of K convex functions, we propose multiple-splitting algorithms for solving them. We propose both basic and accelerated algorithms with O(1/e) and O(1/ 3 ) iteration complexity bounds for obtaining an e-optimal solution. To the best of our knowledge, the complexity results presented in these two chapters are the first ones of this type that have been given for splitting and alternating direction type methods. Numerical results on various applications in sparse and low-rank optimization, including compressed sensing, matrix completion, image deblurring, robust principal component analysis, are reported to demonstrate the efficiency of our methods.

In this paper we consider the general problem of designing the optimal quantum detection for minimum-error quantum state discrimination. Since it is hard to get an explicit solution and we must resort to numerical methods for the general case. We show the problem can be cast into a semidefinite programming(SDP) problem, we use the theory of SDP to solve it. Based on the idea, we propose a method for minimum-error discrimination by formulating the design of optimal quantum detection as a standard SDP problem; furthermore, based on the optimality conditions for SDP, we derive a set of optimality conditions for quantum detection. It is exemplified that this method is simple to implement and can be used to design optimum detection operators for minimum-error discrimination efficiently.

Sum of squares (SOS) optimization has been a powerful and influential addition to the theory of optimization in the past decade. Its reliance on relatively large-scale semidefinite programming, however, has seriously challenged its ability to scale in many practical applications. In this paper, we introduce DSOS and SDSOS optimization as more tractable alternatives to sum of squares optimization that rely instead on linear programming and second order cone programming. These are optimization problems over certain subsets of sum of squares polynomials and positive semidefinite matrices and can be of potential interest in general applications of semidefinite programming where scalability is a limitation.

Sums of squares based approximation algorithms for MAX-SAT

Today’s complex robotic designs comprise in some cases a large number of degrees of freedom, enabling for multi-objective task resolution (e.g., humanoid robots or aerial manipulators). This paper tackles the local stability problem of a hierarchical closed-loop inverse kinematics algorithm for such highly redundant robots. We present a method to guarantee this system stability by performing an online tuning of the closed-loop control gains. We define a semi-definite programming problem (SDP) with these gains as decision variables and a discrete-time Lyapunov stability condition as a linear matrix inequality, constraining the SDP optimization problem and guaranteeing the local stability of the prioritized tasks. To the best of authors’ knowledge, this work represents the first mathematical development of an SDP formulation that introduces these stability conditions for a multi-objective closed-loop inverse kinematic problem for highly redundant robots. The validity of the proposed approach is demonstrated through simulation case studies, including didactic examples and a Matlab toolbox for the benefit of the community.

In this paper, we address the motion parameter estimation problem using time difference of arrival (TDOA) measurements, where a mobile source starts from an initial position with constant velocity. The problem is formulated as three non-convex constrained weighted least squares (CWLS) problems. Owing to their non-convex nature, local convergence may occur when solving them by an iterative algorithm, implying good estimates are not guaranteed. We propose to solve these CWLS problems by applying semidefinite relaxation (SDR) to generate three different semidefinite programming (SDP) problems, namely, the motion unconstrained semidefinite programming (MU-SDP), motion constrained semidefinite programming (MC-SDP), and motion parameter direct semidefinite programming (MPD-SDP). The MU-SDP, MC-SDP, and MPD-SDP methods are then extended to the scenario of unknown propagation speed (PS), which is common in underwater and underground localizations. Specifically, for this scenario, the two-step MU-SDP, two-step MC-SDP, and two-step MPD-SDP methods are designed to keep the relaxed SDP problems tight by introducing the penalty function. Simulation results show that MU-SDP performs worse than MC-SDP and MPD-SDP owing to the ignorance of the motion equation, and MC-SDP and MPD-SDP are able to reach the Cramér-Rao lower bound (CRLB) accuracy.

For wireless multicast channels, it is prospective to provide quality of service (QoS) differentiation at physical layer. In this paper, we propose a rank-two transmit beamforming mechanism to support differentiated services (Diffserv) in multi-group multicasting scenarios based on Alamouti space-time coding (ASTC). In QoS-Diffserv multi-group multicast systems, two kinds of users exist, named high-priority users with guaranteed QoS and low-priority users with limited QoS. First the original QoS-Diffserv issue is formulated as a non-convex quadratically constrained quadratic programming (QCQP) program. With the aid of semi-definite relaxation (SDR) technique, we then relax the QCQP problem to a convex semi-definite programming (SDP) problem, which can be easily handled by basic convex optimization tools. To obtain approximate solutions to the original problem which should meet the rank-two constraints, a Gaussian randomization based process and a power control problem are employed together as the last step. Relationship between the QCQP and SDP problems and computational complexity of the proposed algorithm are analysed. Simulation results demonstrate that the proposed rank-two beamforming algorithm could provide near-optimal transmission power under different QoS constraints.

In this paper, we develop a discretisation algorithm with an adaptive scheme for solving a class of combined semi-infinite and semi-definite programming problems. We show that any sequence of points generated by the algorithm contains a convergent subsequence; and furthermore, each accumulation point is a local optimal solution of the combined semi-infinite and semi-definite programming problem. To illustrate the effectiveness of the algorithm, two specific classes of problems are solved. They are relaxations of quadratically constrained semi-infinite quadratic programming problems and semi-infinite eigenvalue problems.

Semialgebraic computations, i.e., the manipulation of sets and logical conditions defined by polynomial inequalities in real variables, are an essential ”primitive” in the analysis and design of hybrid dynamical systems. Fundamental tasks such as reachability analysis, abstraction verification, and the computation of stability and performance certificates, all use these operations extensively, and can quickly become the computational bottleneck in the design process. Although there is a well-developed body of both basic theory and algorithms for these tasks, the practical performance of most available methods is still far from being satisfactory on real-world problems. While there are several possible causes for this (besides the NP-hardness of the task), a sensible explanation lies in the purely algebraic nature of the usual methods, as well as the insistence on exact (as opposed to approximate or ”relaxed”) solutions. For these reasons, there is a strong interest in the development of efficient techniques for (perhaps restricted) classes of semialgebraic problems. In this talk we review the basic elements and present several new results on the SOS approach to semialgebraic computations, that combines symbolic and numerical techniques from real algebra and convex optimization. Its main defining feature is the use and computation of sum of squares (SOS) decompositions for multivariate polynomials via semidefinite programming. These are extended, using the Positivstellensatz, to structured infeasibility certificates for polynomial equations and inequalities. The developed techniques unify and generalize many well-known existing methods. In particular, we will discuss semialgebraic problems with at most two quantifier alternations (i.e., classical polynomial optimization problems as well as the related games and minimax problems). As an example, we will solve in detail a class of zero-sum two-person games with an infinite number of pure strategies, where the payoff function is a polynomial expression of the actions of the players. Although particular emphasis will be given to the hybrid systems viewpoint, the basic ideas and algorithms, as well as these recent extensions, will be illustrated with examples drawn from a broad range of related domains, including dynamical systems and geometric theorem proving.

Semi-supervised kernel learning methods have been received much more attention in the past few years. Traditional semi-supervised Non-Parametric Kernel Learning (NPKL) methods usually formulate the learning task as a Semi-Definite Programming (SDP) problem, which is very time consuming. Although some fast semi-supervised NPKL methods have been proposed recently, they usually scale very poorly. Furthermore, many semi-supervised NPKL methods are developed based on the manifold assumption. But, such an assumption might be invalid when handling some high-dimensional and sparse data, which has severely negative effect on the performance of learning algorithms. In this paper, we propose a more efficient semi-supervised NPKL method, which can effectively learn a low-rank kernel matrix from must-link and cannot-link constraints. Specially, by virtue of the nonlinear spectral embedded technique based on extreme learning machine (ELM), the proposed method has the ability of coping with data points that do not have a clear manifold structure in a low dimensional space. The proposed method is formulated as a trace ratio optimization problem, which is combined with dimensionality reduction in ELM feature space and aims to find optimal low-rank kernel matrices. The proposed optimization problem can be solved much more efficiently than SDP solvers. Extensive experiments have validated the superior performance of the proposed method compared to state-of-the-art semi-supervised kernel learning methods.