Polynomial Optimization Research Articles

CS-TSSOS: Correlative and Term Sparsity for Large-Scale Polynomial Optimization

This work proposes a new moment-SOS hierarchy, called CS-TSSOS , for solving large-scale sparse polynomial optimization problems. Its novelty is to exploit simultaneously correlative sparsity and term sparsity by combining advantages of two existing frameworks for sparse polynomial optimization. The former is due to Waki et al. [ 40 ] while the latter was initially proposed by Wang et al. [ 42 ] and later exploited in the TSSOS hierarchy [ 46 , 47 ]. In doing so we obtain CS-TSSOS—a two-level hierarchy of semidefinite programming relaxations with (i) the crucial property to involve blocks of SDP matrices and (ii) the guarantee of convergence to the global optimum under certain conditions. We demonstrate its efficiency and scalability on several large-scale instances of the celebrated Max-Cut problem and the important industrial optimal power flow problem, involving up to six thousand variables and tens of thousands of constraints.

Exploiting Constant Trace Property in Large-scale Polynomial Optimization

We prove that every semidefinite moment relaxation of a polynomial optimization problem (POP) with a ball constraint can be reformulated as a semidefinite program involving a matrix with constant trace property (CTP). As a result, such moment relaxations can be solved efficiently by first-order methods that exploit CTP, e.g., the conditional gradient-based augmented Lagrangian method. We also extend this CTP-exploiting framework to large-scale POPs with different sparsity structures. The efficiency and scalability of our framework are illustrated on some moment relaxations for various randomly generated POPs, especially second-order moment relaxations for quadratically constrained quadratic programs.

Algebraic Perspectives on Signomial Optimization

Signomials are obtained by generalizing polynomials to allow for arbitrary real exponents. This generalization offers great expressive power but has historically sacrificed the organizing principle of “degree” that is central to polynomial optimization theory. We reclaim that principle here through the concept of signomial rings, which we use to derive complete convex relaxation hierarchies of upper and lower bounds for signomial optimization via sums of arithmetic-geometric exponentials (SAGE) nonnegativity certificates. The Positivstellensatz underlying the lower bounds relies on the concept of conditional SAGE and removes regularity conditions required by earlier works, such as convexity of the feasible set or Archimedeanity of its representing signomial inequalities. Through worked examples we illustrate the practicality of this hierarchy in areas such as chemical reaction network theory and chemical engineering. These examples include comparisons to direct global solvers (e.g., BARON and ANTIGONE) and the Lasserre hierarchy (where appropriate). The completeness of our hierarchy of upper bounds follows from a generic construction whereby a Positivstellensatz for signomial nonnegativity over a compact set provides for arbitrarily strong outer approximations of the corresponding cone of nonnegative signomials. While working toward that result, we prove basic facts on the existence and uniqueness of solutions to signomial moment problems.

An inexact projected gradient method with rounding and lifting by nonlinear programming for solving rank-one semidefinite relaxation of polynomial optimization

We consider solving high-order and tight semidefinite programming (SDP) relaxations of nonconvex polynomial optimization problems (POPs) that often admit degenerate rank-one optimal solutions. Instead of solving the SDP alone, we propose a new algorithmic framework that blends local search using the nonconvex POP into global descent using the convex SDP. In particular, we first design a globally convergent inexact projected gradient method (iPGM) for solving the SDP that serves as the backbone of our framework. We then accelerate iPGM by taking long, but safeguarded, rank-one steps generated by fast nonlinear programming algorithms. We prove that the new framework is still globally convergent for solving the SDP. To solve the iPGM subproblem of projecting a given point onto the feasible set of the SDP, we design a two-phase algorithm with phase one using a symmetric Gauss–Seidel based accelerated proximal gradient method (sGS-APG) to generate a good initial point, and phase two using a modified limited-memory BFGS (L-BFGS) method to obtain an accurate solution. We analyze the convergence for both phases and establish a novel global convergence result for the modified L-BFGS that does not require the objective function to be twice continuously differentiable. We conduct numerical experiments for solving second-order SDP relaxations arising from a diverse set of POPs. Our framework demonstrates state-of-the-art efficiency, scalability, and robustness in solving degenerate SDPs to high accuracy, even in the presence of millions of equality constraints.

Convergence of an SDP hierarchy and optimality of robust convex polynomial optimization problems

Convergence of an SDP hierarchy and optimality of robust convex polynomial optimization problems

PoBO: A Polynomial Bounding Method for Chance-Constrained Yield-Aware Optimization of Photonic ICs

Conventional yield optimization algorithms try to maximize the success rate of a circuit under process variations. These methods often obtain a high yield but reach a design performance that is far from the optimal value. This article investigates an alternative yield-aware optimization for photonic ICs: we will optimize the circuit design performance while ensuring a high yield requirement. This problem was recently formulated as a chance-constrained optimization, and the chance constraint was converted to a stronger constraint with statistical moments. Such a conversion reduces the feasible set and sometimes leads to an over-conservative design. To address this fundamental challenge, this article proposes a carefully designed polynomial function, called optimal polynomial kinship function, to bound the chance constraint more accurately. We modify existing kinship functions via relaxing the independence and convexity requirements, which fits our more general uncertainty modeling and tightens the bounding functions. The proposed method enables a global optimum search for the design variables via polynomial optimization. We validate this method with a synthetic function and two photonic IC design benchmarks, showing that our method can obtain better design performance while meeting a prespecified yield requirement. Many other advanced problems of yield-aware optimization and more general safety-critical design/control can be solved based on this work in the future.

Ant Based System: Overview, Modifications and Applications from 1992 to 2022

The complexity of real-world problems motivated the researchers to innovate efficient problem-solving techniques. Generally, natural-inspired, bio-inspired, metaheuristics-based evolutionary computation, and swarm intelligence algorithms have been frequently used for solving complex, real-world optimization and non-deterministic polynomial hard (NP-Hard) problems due to their ability to adjust to a variety of conditions. This paper shows an overview of swarm-based algorithms based on Ant behavior. The first algorithm that inspired Ant behavior in the search for food sources was developed in 1992 and tested in solving the TSP problem. Ant Colony Optimization (ACO) is a metaheuristic inspired by some Ant species' pheromone trail laying following behavior. Artificial Ant in ACO is a stochastic solution construction process that uses (artificial) pheromone information that is modified depending on the Ant’s search experience which is possibly accessible in heuristic information to generate solutions for problems. Notable research has been gained since the proposal of the Ant system. These contribute to the creation of high-performance algorithmic variants of a generic algorithmic framework for ACO, its successful application to a wide range of computing difficult problems, and the theoretical understanding of its properties.

Optimality conditions for homogeneous polynomial optimization on the unit sphere

In this note, we prove that for homogeneous polynomial optimization on the sphere, if the objective f is generic in the input space, all feasible points satisfying the first order and second order necessary optimality conditions are local minimizers, which addresses an issue raised in the recent work by Lasserre (Optimization Letters, 2021). As a corollary, this implies that Lasserre’s hierarchy has finite convergence when f is generic.

Dual Certificates and Efficient Rational Sum-of-Squares Decompositions for Polynomial Optimization over Compact Sets

We study the problem of computing weighted sum-of-squares (WSOS) certificates for positive polynomials over a compact semialgebraic set. Building on the theory of interior-point methods for convex optimization, we introduce the concept of dual certificates, which allows us to interpret vectors from the dual of the sum-of-squares cone as rigorous nonnegativity certificates of a WSOS polynomial. Whereas conventional WSOS certificates are alternative representations of the polynomials they certify, dual certificates are distinct from the certified polynomials; moreover, each dual certificate certifies a full-dimensional convex cone of WSOS polynomials. For a theoretical application, we give a short new proof of Powers's theorems on the existence of rational WSOS certificates of positive polynomials. For a computational application, we show that exact WSOS certificates can be constructed from numerically computed dual certificates at little additional cost, without any rounding or projection steps applied to the numerical certificates. We also present an algorithm for computing the optimal WSOS lower bound of a given polynomial along with a rational dual certificate, with a polynomial-time computational cost per iteration and linear rate of convergence.

Sums of Separable and Quadratic Polynomials

We study separable plus quadratic (SPQ) polynomials, that is, polynomials that are the sum of univariate polynomials in different variables and a quadratic polynomial. Motivated by the fact that nonnegative separable and nonnegative quadratic polynomials are sums of squares, we study whether nonnegative SPQ polynomials are (i) the sum of a nonnegative separable and a nonnegative quadratic polynomial and (ii) a sum of squares. We establish that the answer to question (i) is positive for univariate plus quadratic polynomials and for convex SPQ polynomials but negative already for bivariate quartic SPQ polynomials. We use our decomposition result for convex SPQ polynomials to show that convex SPQ polynomial optimization problems can be solved by “small” semidefinite programs. For question (ii), we provide a complete characterization of the answer based on the degree and the number of variables of the SPQ polynomial. We also prove that testing nonnegativity of SPQ polynomials is NP-hard when the degree is at least four. We end by presenting applications of SPQ polynomials to upper bounding sparsity of solutions to linear programs, polynomial regression problems in statistics, and a generalization of Newton’s method that incorporates separable higher order derivative information. Funding: This work was partially supported by an AFOSR MURI award, the Defense Advanced Research Projects Agency Young Faculty Award, the Princeton School of Engineering and Applied Sciences Innovation Award, the National Science Foundation Faculty Early Career Development Program Award, the Google Faculty Award, and the Sloan Fellowship.

Optimal Transmission Power and Controller Design for Networked Control Systems Under State-Dependent Markovian Channels

This article considers a codesign problem for industrial networked control systems to ensure both stability and efficiency properties of such systems. This problem is particularly challenging due to the fact that wireless communications in industrial environments are not only subject to shadow fading, but also stochastically correlated with their surrounding environments. This article first introduces a novel state-dependent Markov channel (SD-MC) model that explicitly captures the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">state-dependent features</i> of industrial wireless communication systems by defining the proposed model’s transition probabilities as a function of both environments’ states and transmission power. Under the SD-MC model, sufficient conditions on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Maximum Allowable Transmission Interval</i> are presented to ensure both <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">asymptotic stability in expectation</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">almost sure asymptotic stability</i> properties of a nonlinear control system with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">state-dependent fading channels</i> . Based on these stability conditions, the codesign problem is then formulated as a constrained polynomial optimization problem (CPOP), which can be efficiently solved using semidefinite programming methods for the case of a two-state SD-MC model. The solutions to such a CPOP represent optimal control and power strategies that optimize the average expected joint costs in an infinite time horizon while respecting the stability constraints. For a general SD-MC model, this article further shows that suboptimal solutions can be obtained from linear programming formulations of the considered CPOP. Simulation results are given to illustrate the efficacy of the proposed codesign scheme.

Computational advances in polynomial optimization: RAPOSa, a freely available global solver

In this paper we introduce RAPOSa, a global optimization solver specifically designed for (continuous) polynomial programming problems with box-constrained variables. Written entirely in C++, RAPOSa is based on the Reformulation-Linearization (Sherali and Tuncbilek in J Glob Optim 103:225–249, 1992). We present a description of the main characteristics of RAPOSa along with a thorough analysis of the impact on its performance of various enhancements discussed in the literature, such as bound tightening and SDP cuts. We also present a comparative study with three of the main state-of-the-art global optimization solvers: BARON, Couenne and SCIP.

A disintegration of the Christoffel function

We show that the Christoffel function (CF) factorizes (or can be disintegrated) as the product of two Christoffel functions, one associated with the marginal and the another related to the conditional distribution, in the spirit of “the CF of the disintegration is the disintegration of the CFs”. In the proof one uses an apparently overlooked property (but interesting in its own) which states that any sum-of-squares polynomial is the Christoffel function of some linear form (with a representing measure in the univariate case). The same is true for the convex cone of polynomials that are positive on a basic semi-algebraic set. This interpretation of the CF establishes another bridge between polynomials optimization and orthogonal polynomials.

The analytic connectivity in uniform hypergraphs: Properties and computation

AbstractThe analytic connectivity (AC), defined via solving a series of constrained polynomial optimization problems, serves as a measure of connectivity in hypergraphs. How to compute such a quantity efficiently is important in practice and of theoretical challenge as well due to the non‐convex and combinatorial features in its definition. In this article, we first perform a careful analysis of several widely used structured hypergraphs in terms of their properties and heuristic upper bounds of ACs. We then present an affine‐scaling method to compute some upper bounds of ACs for uniform hypergraphs. To testify the tightness of the obtained upper bounds, two possible approaches via the Pólya theorem and semidefinite programming respectively are also proposed to verify the lower bounds generated by the obtained upper bounds minus a small gap. Numerical experiments on synthetic datasets are reported to demonstrate the efficiency of our proposed method. Further, we apply our method in hypergraphs constructed from social networks and text analysis to detect the network connectivity and rank the keywords, respectively.

Efficient Global Solutions to Single-Input Optimal Control Problems via Approximation by Sum-of-Squares Polynomials

Optimal control problems are prevalent in model-based control, state and parameter estimation, and experimental design for complex dynamical systems. An approach for obtaining solutions to these problems is based on the notion of parsimonious input parameterization and comprises two tasks: the enumeration of arc sequences followed by the computation of optimal values of a small number of decision variables for each sequence. This article proposes an efficient global solution method for single-input optimal control problems for nonlinear dynamical systems with a potentially large number of states or complex dynamics via sum-of-squares polynomials and parallel computing. The method approximates the problem for a given arc sequence as a polynomial optimization problem that can be efficiently solved to global optimality via semidefinite programming. It is established that the difference between the cost obtained by the proposed method and the globally optimal cost of the original problem is bounded and depends on the polynomial approximation error. The method is illustrated by simulation examples of a reaction system and a rocket.

Exactness of Parrilo’s Conic Approximations for Copositive Matrices and Associated Low Order Bounds for the Stability Number of a Graph

De Klerk and Pasechnik introduced in 2002 semidefinite bounds [Formula: see text] for the stability number [Formula: see text] of a graph G and conjectured their exactness at order [Formula: see text]. These bounds rely on the conic approximations [Formula: see text] introduced by Parrilo in 2000 for the copositive cone [Formula: see text]. A difficulty in the convergence analysis of the bounds is the bad behavior of Parrilo’s cones under adding a zero row/column: when applied to a matrix not in [Formula: see text] this gives a matrix that does not lie in any of Parrilo’s cones, thereby showing that their union is a strict subset of the copositive cone for any [Formula: see text]. We investigate the graphs for which the bound of order r≤1 is exact: we algorithmically reduce testing exactness of [Formula: see text] to acritical graphs, we characterize the critical graphs with [Formula: see text] exact, and we exhibit graphs for which exactness of [Formula: see text] is not preserved under adding an isolated node. This disproves a conjecture posed by Gvozdenović and Laurent in 2007, which, if true, would have implied the above conjecture by de Klerk and Pasechnik. Funding: This work is supported by the European Union’s Framework Programme for Research and Innovation Horizon 2020 under the Marie Skłodowska-Curie Actions Grant Agreement Polynomial Optimization, Efficiency through Moments and Algebra [Grant 813211].

SOS-Based Nonlinear Observer Design for Simultaneous State and Disturbance Estimation Designed for a PMSM Model

In this study, a type of nonlinear observer design is studied for a class of nonlinear systems. For the construction of the nonlinear observer, SOS-based optimization tools are utilized, which for some nonlinear dynamical systems have the advantage of transforming the problem into a more tractable one. The general problem of nonlinear observer design is translated into an SOS polynomial optimization which can be turned into an SDP problem. For a study problem, simultaneous state and disturbance estimation is considered, a cascaded nonlinear observer using a certain parameterization is constructed, and computation techniques are discussed. Cascade nonlinear observer structure is a design strategy that decomposes the problem into its components resulting in dimension reduction. In this paper, SOS-based methods using the cascade design technique are represented, and a simultaneous state and disturbance signal online estimation algorithm is constructed. The method with its smaller components is given in detail, the efficacy of the method is demonstrated by means of numerical simulations performed in MATLAB, and the observer is designed using numerical optimization tools YALMIP, MOSEK, and PENLAB.

Homogenization for polynomial optimization with unbounded sets

This paper considers polynomial optimization with unbounded sets. We give a homogenization formulation and propose a hierarchy of Moment-SOS relaxations to solve it. Under the assumptions that the feasible set is closed at infinity and the ideal of homogenized equality constraining polynomials is real radical, we show that this hierarchy of Moment-SOS relaxations has finite convergence, if some optimality conditions (i.e., the linear independence constraint qualification, strict complementarity and second order sufficient conditions) hold at every minimizer, including the one at infinity. Moreover, we prove extended versions of Putinar-Vasilescu type Positivstellensatz for polynomials that are nonnegative on unbounded sets. The classical Moment-SOS hierarchy with denominators is also studied. In particular, we give a positive answer to a conjecture of Mai, Lasserre and Magron in their recent work.

Partial Lasserre relaxation for sparse Max-Cut

A common approach to solve or find bounds of polynomial optimization problems like Max-Cut is to use the first level of the Lasserre hierarchy. Higher levels of the Lasserre hierarchy provide tighter bounds, but solving these relaxations is usually computationally intractable. We propose to strengthen the first level relaxation for sparse Max-Cut problems using constraints from the second order Lasserre hierarchy. We explore a variety of approaches for adding a subset of the positive semidefinite constraints of the second order sparse relaxation obtained by using the maximum cliques of the graph’s chordal extension. We apply this idea to sparse graphs of different sizes and densities, and provide evidence of its strengths and limitations when compared to the state-of-the-art Max-Cut solver BiqCrunch and the alternative sparse relaxation CS-TSSOS.

Comparative study for multi-variable regression methods based on Laguerre polynomial and manifolds optimization

PurposeThe purpose of this paper is to overcome the application limitations of other multi-variable regression based on polynomials due to the huge computation room and time cost.Design/methodology/approachIn this paper, based on the idea of feature selection and cascaded regression, two strategies including Laguerre polynomials and manifolds optimization are proposed to enhance the accuracy of multi-variable regression. Laguerre polynomials were combined with the genetic algorithm to enhance the capacity of polynomials approximation and the manifolds optimization method was introduced to solve the co-related optimization problem.FindingsTwo multi-variable Laguerre polynomials regression methods are designed. Firstly, Laguerre polynomials are combined with feature selection method. Secondly, manifolds component analysis is adopted in cascaded Laguerre polynomials regression method. Two methods are brought to enhance the accuracy of multi-variable regression method.Research limitations/implicationsWith the increasing number of variables in regression problem, the stable accuracy performance might not be kept by using manifold-based optimization method. Moreover, the methods mentioned in this paper are not suitable for the classification problem.Originality/valueExperiments are conducted on three types of datasets to evaluate the performance of the proposed regression methods. The best accuracy was achieved by the combination of cascade, manifold optimization and Chebyshev polynomials, which implies that the manifolds optimization has stronger contribution than the genetic algorithm and Laguerre polynomials.