Sum Of Squares Polynomials Research Articles

Approximation and Interpolation of Singular Measures by Trigonometric Polynomials

AbstractComplex valued measures of finite total variation are a powerful signal model in many applications. Restricting to the d-dimensional torus, finitely supported measures can be exactly recovered from their trigonometric moments up to some order if this order is large enough. Here, we consider the approximation of general measures, e.g., supported on a curve, by trigonometric polynomials of fixed degree with respect to the 1-Wasserstein distance. We prove sharp lower bounds for their best approximation and (almost) matching upper bounds for effectively computable approximations when the trigonometric moments of the measure are known. A second class of sum of squares polynomials is shown to interpolate the indicator function on the support of the measure and to converge to zero outside.

Shape-Constrained Regression Using Sum of Squares Polynomials

Shape-Constrained Regression Using Algebraic Techniques How can one fit a multivariate polynomial to data points in such a way that this polynomial is guaranteed to have certain shape constraints, such as convexity or monotonicity in some of its variables? In “Shape-constrained regression using sum-of-squares polynomials,” M. Curmei and G. Hall propose a hierarchy of semidefinite programs to address this problem. They show that polynomial functions that are optimal to any fixed level of our hierarchy form a consistent estimator of the underlying shape-constrained function, which generates the data points. As a by-product of the proof, they establish that sum-of-squares-convex polynomials are dense in the set of polynomials that are convex over an arbitrary box. They further demonstrate the performance of the regressor for the problem of computing optimal transport maps in a color transfer task and that of estimating the optimal value function of a conic program. A real-time application of the latter problem to inventory management contract negotiation is presented.

Convergences for robust bilevel polynomial programmes with applications

In this paper, we consider a bilevel polynomial optimization problem, where the constraint functions of both the upper-level and lower-level problems involve uncertain parameters. We employ the deterministic robust optimization approach to examine the bilevel polynomial optimization problem under data uncertainties by providing lower bound approximations and convergences of sum-of-squares (SOS) relaxations for the robust bilevel polynomial optimization problem. More precisely, we show that under the convexity of the lower-level problem and either the boundedness of the feasible set or the coercivity of the objective function, the global optimal values of SOS relaxation problems are lower bounds of the global optimal value of the robust bilevel polynomial problem and they converge to this global optimal value when the degrees of SOS polynomials in the relaxation problems tend to infinity. Moreover, an application to an electric vehicle charging scheduling problem with renewable energy sources demonstrates that using the proposed SOS relaxation schemes, we obtain more stable optimal values than applying a direct solution approach as the SOS relaxations are capable of solving these models involving data uncertainties in dynamic charging price and weather conditions.

Passive Scalar Function Approximation Using SOS Polynomials

Signal and power integrity design in the time domain requires equivalent circuit models for interconnects and packages, whose descriptions may only be available as tabulated impedance or admittance parameters. Accurate models for these components should maintain their physical properties including causality, stability, and passivity. Even though a heuristic approach, pole flipping (i.e., changing the sign of the real part of an unstable pole) has proven to be sufficiently accurate for many applications, resulting in models with ensured causality and stability. In this article, we address the problem of generating passive scalar models, such as driving point impedances or admittances, based on an existing causal, stable, but nonpassive model. Our approach is based on using sum-of-squares (SOS) polynomials and results in a convex optimization problem, hence a global optimum is obtained. We obtain approximations through two distinct SOS algorithms: a coefficient and a sampling-based method. We also present a simple passivity test for such functions based on calculating the roots of numerator and denominator polynomials, which provides an intuitive link among existing approaches based on solving an eigenvalue problem.

On the product of the singular values of a binary tensor

A real binary tensor consists of $2^d$ real entries arranged into hypercube format $2^{\times d}$. For $d=2$, a real binary tensor is a $2\times 2$ matrix with two singular values. Their product is the determinant. We generalize this formula for any $d\ge 2$. Given a partition $\mu\vdash d$ and a $\mu$-symmetric real binary tensor $t$, we study the distance function from $t$ to the variety $X_{\mu,\mathbb{R}}$ of $\mu$-symmetric real binary tensors of rank one. The study of the local minima of this function is related to the computation of the singular values of $t$. Denoting with $X_\mu$ the complexification of $X_{\mu,\mathbb{R}}$, the Euclidean Distance polynomial $\mathrm{EDpoly}_{X_\mu^\vee,t}(\epsilon^2)$ of the dual variety of $X_\mu$ at $t$ has among its roots the singular values of $t$. On one hand, the lowest coefficient of $\mathrm{EDpoly}_{X_\mu^\vee,t}(\epsilon^2)$ is the square of the $\mu$-discriminant of $t$ times a product of sum of squares polynomials. On the other hand, we describe the variety of $\mu$-symmetric binary tensors that do not admit the maximum number of singular values, counted with multiplicity. Finally, we compute symbolically all the coefficients of $\mathrm{EDpoly}_{X_\mu^\vee,t}(\epsilon^2)$ for tensors of format $2\times 2\times 2$.

Stable Rank-One Matrix Completion is Solved by the Level 2 Lasserre Relaxation

This paper studies the problem of deterministic rank-one matrix completion. It is known that the simplest semidefinite programming relaxation, involving minimization of the nuclear norm, does not in general return the solution for this problem. In this paper, we show that in every instance where the problem has a unique solution, one can provably recover the original matrix through the level 2 Lasserre relaxation with minimization of the trace norm. We further show that the solution of the proposed semidefinite program is Lipschitz stable with respect to perturbations of the observed entries, unlike more basic algorithms such as nonlinear propagation or ridge regression. Our proof is based on recursively building a certificate of optimality corresponding to a dual sum-of-squares (SoS) polynomial. This SoS polynomial is built from the polynomial ideal generated by the completion constraints and the monomials provided by the minimization of the trace. The proposed relaxation fits in the framework of the Lasserre hierarchy, albeit with the key addition of the trace objective function. Finally, we show how to represent and manipulate the moment tensor in favorable complexity by means of a hierarchical low-rank factorization.

Computation of Sum of Squares Polynomials from Data Points

We propose an iterative algorithm for the numerical computation of sums of squares of polynomials approximating given data at prescribed interpolation points. The method is based on the definition ...

On the Construction of Converging Hierarchies for Polynomial Optimization Based on Certificates of Global Positivity

In recent years, techniques based on convex optimization and real algebra that produce converging hierarchies of lower bounds for polynomial minimization problems have gained much popularity. At their heart, these hierarchies rely crucially on Positivstellensätze from the late 20th century (e.g., due to Stengle, Putinar, or Schmüdgen) that certify positivity of a polynomial on an arbitrary closed basic semialgebraic set. In this paper, we show that such hierarchies could in fact be designed from much more limited Positivstellensätze dating back to the early 20th century that only certify positivity of a polynomial globally. More precisely, we show that any inner approximation to the cone of positive homogeneous polynomials that is arbitrarily tight can be turned into a converging hierarchy of lower bounds for general polynomial minimization problems with compact feasible sets. This in particular leads to a semidefinite programming–based hierarchy that relies solely on Artin’s solution to Hilbert’s 17th problem. We also use a classical result from Pólya on global positivity of even forms to construct an “optimization-free” converging hierarchy for general polynomial minimization problems with compact feasible sets. This hierarchy requires only polynomial multiplication and checking nonnegativity of coefficients of certain fixed polynomials. As a corollary, we obtain new linear programming–based and second-order cone programming–based hierarchies for polynomial minimization problems that rely on the recently introduced concepts of diagonally dominant sum of squares and scaled diagonally dominant sum of squares polynomials. We remark that the scope of this paper is theoretical at this stage, as our hierarchies—though they involve at most two sum of squares constraints or only elementary arithmetic at each level—require the use of bisection and increase the number of variables (respectively, the degree) of the problem by the number of inequality constraints plus three (respectively, by a factor of two).

Alternative SDP and SOCP approximations for polynomial optimization

In theory, hierarchies of semidefinite programming (SDP) relaxations based on sum of squares (SOS) polynomials have been shown to provide arbitrarily close approximations for a general polynomial optimization problem (POP). However, due to the computational challenge of solving SDPs, it becomes difficult to use SDP hierarchies for large-scale problems. To address this, hierarchies of second-order cone programming (SOCP) relaxations resulting from a restriction of the SOS polynomial condition have been recently proposed to approximate POPs. Here, we consider alternative ways to use this SOCP restrictions of the SOS condition. In particular, we show that SOCP hierarchies can be effectively used to strengthen hierarchies of linear programming (LP) relaxations for POPs. Specifically, we show that this solution approach is substantially more effective in finding solutions of certain POPs for which the more common hierarchies of SDP relaxations are known to perform poorly. Furthermore, when the feasible set of the POP is compact, these SOCP hierarchies converge to the POP's optimal value.

Dwell time-based stabilisation of switched linear delay systems using clock-dependent Lyapunov–Krasovskii functionals

ABSTRACTDwell time stability conditions of the switched delay systems are derived using multiple clock-dependent Lyapunov–Krasovskii functionals. The corresponding conditions are approximated by both using piecewise linear functions and sum of squares polynomials. The upper bound of the dwell time is minimised using a combination of a bisection and a golden section search algorithm. Using the results obtained in the stability case, synthesis of dwell time minimiser controllers are presented. Some numerical examples are given to illustrate effectiveness of the proposed method, and its performance is compared with the existing approaches. The resulting values of the dwell time via the proposed technique show that the novel approach outperforms the previous ones.

Observer‐based non‐linear H ∞ attitude control for a flexible satellite

This study focuses on the observer-based non-linear H ∞ (OBNH) attitude control for a flexible satellite with external disturbances and local state-region constraints. By considering the structural features of the attitude system and applying Lyapunov stability theory, the solvable conditions of the OBNH attitude control problem are provided. An advantage of this study is that the state-feedback controller and the reduced-order state observer can be designed independently, thus significantly reducing the complexity of the attitude control algorithm. Furthermore, based on the polynomial sum of squares (SOSs) and the generalised S-procedure techniques, the above problem can be transformed into a convex optimisation problem with SOS constraints. This conversion can effectively overcome the computational difficulties widely present in the control problem of non-linear systems. Finally, a numerical example is given to illustrate the effectiveness and feasibility of the proposed approach.

On globally diffeomorphic polynomial maps via Newton polytopes and circuit numbers

In this article we analyze the global diffeomorphism property of polynomial maps $$F:\mathbb {R}^n\rightarrow \mathbb {R}^n$$ by studying the properties of the Newton polytopes at infinity corresponding to the sum of squares polynomials $$\Vert F\Vert _2^2$$ . This allows us to identify a class of polynomial maps F for which their global diffeomorphism property on $$\mathbb {R}^n$$ is equivalent to their Jacobian determinant $$\det JF$$ vanishing nowhere on $$\mathbb {R}^n$$ . In other words, we identify a class of polynomial maps for which the Real Jacobian Conjecture, which was proven to be false in general, still holds.

New LMI conditions for admissibility analysis of time-varying descriptor systems

This paper presents new admissibility analysis conditions for discrete-time descriptor systems. Firstly, it proposes an extension, expressed as necessary and sufficient admissibility conditions, of admissibility criteria developed in the literature for linear time-invariant systems in order to incorporate the time-varying case. This extension exploits time-varying slack variables and uses the linear matrix inequality (LMI) formulation. Then, this extension is applied to the case of polytopic descriptor systems with bounded time-varying parameters. The use of sum of squares polynomials allows to eliminate the criteria parametric dependence and to provide new admissibility analysis conditions expressed as a LMIs feasibility problem. This new admissibility analysis result uses parameter-dependent slack variables and may be less conservative than the results already proposed in the literature as sustained by our numerical evaluation.

Fiber Orientation Distribution Estimation Using a Peaceman--Rachford Splitting Method

In diffusion-weighted magnetic resonance imaging, the estimation of the orientations of multiple nerve fibers in each voxel (the fiber orientation distribution (FOD)) is a critical issue for exploring the connection of cerebral tissue. In this paper, we establish a convex semidefinite programming (CSDP) model for the FOD estimation. One feature of the new model is that it can ensure the statistical meaning of FOD since as a probability density function, FOD must be nonnegative and have a unit mass. To construct such a statistically meaningful FOD, we consider its approximation by a sum of squares (SOS) polynomial and impose the unit-mass by a linear constraint. Another feature of the new model is that it introduces a new regularization based on the sparsity of nerve fibers. Due to the sparsity of the orientations of nerve fibers in cerebral white matter, a heuristic regularization is raised, which is inspired by the Z-eigenvalue of a symmetric tensor that closely relates to the SOS polynomial. To solve th...

Observer-based robust stabilization for a class of uncertain polynomial systems

This paper focuses on the observer-based robust stabilization problem for a class of polynomial systems with norm-bounded time-varying uncertainties. The structural features of such systems guarantee the fulfilment of the separation principle between the reduced-order observer and the state feedback. The existence conditions of the observer-based robust stabilization controller are obtained by using Lyapunov stability theory. Furthermore, based on the polynomial sum of squares (SOS) theory, the above conditions are transformed into the corresponding SOS convex optimization constraints, for the avoidance of computing difficulties that exist widely in the control of non-linear systems. Finally, two numerical examples are given to illustrate the feasibility and effectiveness of the proposed approach.

Polynomial sum of squares in fluid dynamics: a review with a look ahead.

The first part of this paper reviews the application of the sum-of-squares-of-polynomials technique to the problem of global stability of fluid flows. It describes the known approaches and the latest results, in particular, obtaining for a version of the rotating Couette flow a better stability range than the range given by the classic energy stability method. The second part of this paper describes new results and ideas, including a new method of obtaining bounds for time-averaged flow parameters illustrated with a model problem and a method of obtaining approximate bounds that are insensitive to unstable steady states and periodic orbits. It is proposed to use the bound on the energy dissipation rate as the cost functional in the design of flow control aimed at reducing turbulent drag.

Rational Lyapunov functions for estimating and controlling the robust domain of attraction

This paper addresses the estimation and control of the robust domain of attraction (RDA) of equilibrium points through rational Lyapunov functions (LFs) and sum of squares (SOS) techniques. Specifically, continuous-time uncertain polynomial systems are considered, where the uncertainty is represented by a vector that affects polynomially the system and is constrained into a semialgebraic set. The estimation problem consists of computing the largest estimate of the RDA (LERDA) provided by a given rational LF. The control problem consists of computing a polynomial static output controller of given degree for maximizing such a LERDA. In particular, the paper shows that the computation of the best lower bound of the LERDA for chosen degrees of the SOS polynomials, which requires the solution of a nonconvex optimization problem with bilinear matrix inequalities (BMIs), can be reformulated as a quasi-convex optimization problem under some conditions. Moreover, the paper provides a necessary and sufficient condition for establishing tightness of this lower bound. Lastly, the paper discusses the search for optimal rational LFs using the proposed strategy.

The Pythagoras number of real sum of squares polynomials and sum of square magnitudes of polynomials

In this paper, we conjecture a formula for the value of the Pythagoras number for real multivariate sum of squares polynomials as a function of the (total or coordinate) degree and the number of variables. The conjecture is based on the comparison between the number of parameters and the number of conditions for a corresponding low-rank representation. This is then numerically verified for a number of examples. Additionally, we discuss the Pythagoras number of (complex) multivariate Laurent polynomials that are sum of square magnitudes of polynomials on the \(n\)-torus. For both types of polynomials, we also propose an algorithm to numerically compute the Pythagoras number and give some numerical illustrations.

Sum of Squares Based Input-to-State Stability Analysis of Polynomial Nonlinear Systems

Input-to-state stability (ISS) is an advanced concept of stability for nonlinear systems in the sense that the input to the system is taken into account. This paper proposes a numerically tractable condition to analyze ISS for polynomial nonlinear systems. The condition is formulated as a sum of squares (SOS) problem that is a feasibility problem including SOS polynomials. A solution of the problem is obtained by solving a semidefinite programming problem. The key technique is to formulate class K∞ functions, which have the inverse map on real nonnegative region, as polynomials. This paper also gives numerically tractable ISS conditions for the cascade connection of subsystems and for the feedback interconnected system. A few examples are shown to illustrate our analysis conditions.

Exact certification in global polynomial optimization via sums-of-squares of rational functions with rational coefficients

We present a hybrid symbolic-numeric algorithm for certifying a polynomial or rational function with rational coefficients to be non-negative for all real values of the variables by computing a representation for it as a fraction of two polynomial sum-of-squares (SOS) with rational coefficients. Our new approach turns the earlier methods by Peyrl and Parrilo at SNC’07 and ours at ISSAC’08 both based on polynomial SOS, which do not always exist, into a universal algorithm for all inputs via Artin’s theorem. Furthermore, we scrutinize the all-important process of converting the numerical SOS numerators and denominators produced by block semidefinite programming into an exact rational identity. We improve on our own Newton iteration-based high precision refinement algorithm by compressing the initial Gram matrices and by deploying rational vector recovery aside from orthogonal projection. We successfully demonstrate our algorithm on (1) various exceptional SOS problems with necessary polynomial denominators from the literature and on (2) very large (thousands of variables) SOS lower bound certificates for Rump’s model problem (up to n = 18 , factor degree = 17 ).