Related papers: Bernstein Polynomial Relaxations for Polynomial Op…
Polynomial optimization problems over binary variables can be expressed as integer programs using a linearization with extra monomials in addition to those arising in the given polynomial. We characterize when such a linearization yields an…
This work considers polynomial optimization problems where the objective admits a low-rank canonical polyadic tensor decomposition. We introduce LRPOP (low-rank polynomial optimization), a new hierarchy of semidefinite programming…
In this paper, we examine linear programming (LP) based relaxations for synthesizing polynomial Lyapunov functions to prove the stability of polynomial ODEs. A common approach to Lyapunov function synthesis starts from a desired parametric…
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…
A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In…
Polynomial optimization problems (POPs) can be reformulated as geometric convex conic programs, as shown by Kim, Kojima, and Toh (SIOPT 30:1251-1273, 2020), though such formulations remain NP-hard. In this work, we prove that several…
In this paper, we investigate the problem of finding tight linear lower bounding functions for multivariate polynomials over boxes. These functions are obtained by the expansion of polynomials into Bernstein form and using the linear least…
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…
Completely positive (CP) tensors, which correspond to a generalization of CP matrices, allow to reformulate or approximate a general polynomial optimization problem (POP) with a conic optimization problem over the cone of CP tensors.…
We study T-semidefinite programming (SDP) relaxation for constrained polynomial optimization problems (POPs). T-SDP relaxation for unconstrained POPs was introduced by Zheng, Huang and Hu in 2022. In this work, we propose a T-SDP relaxation…
Polynomial optimization encompasses a broad class of problems in which both the objective function and constraints are polynomial functions of the decision variables. In recent years, a substantial body of research has focused on…
The problem of optimizing over the cone of nonnegative polynomials is a fundamental problem in computational mathematics, with applications to polynomial optimization, control, machine learning, game theory, and combinatorics, among others.…
In this paper, we consider a bilevel polynomial optimization problem where the objective and the constraint functions of both the upper and the lower level problems are polynomials. We present methods for finding its global minimizers and…
We consider polynomial optimization problems (POP) on a semialgebraic set contained in the nonnegative orthant (every POP on a compact set can be put in this format by a simple translation of the origin). Such a POP can be converted to an…
Bernstein polynomial approximation to a continuous function has a slower rate of convergence as compared to other approximation methods. "The fact seems to have precluded any numerical application of Bernstein polynomials from having been…
We present a hierarchy of tractable relaxations to obtain lower bounds on the minimum value of a polynomial over a constraint set defined by polynomial equations. In contrast to previous convex relaxation techniques for this problem, our…
This paper proposes tight semidefinite relaxations for polynomial optimization. The optimality conditions are investigated. We show that generally Lagrange multipliers can be expressed as polynomial functions in decision variables over the…
We study how to solve semidefinite programming relaxations for large scale polynomial optimization. When interior-point methods are used, typically only small or moderately large problems could be solved. This paper studies regularization…
We consider the general polynomial optimization problem $P: f^*=\min \{f(x)\,:\,x\in K\}$ where $K$ is a compact basic semi-algebraic set. We first show that the standard Lagrangian relaxation yields a lower bound as close as desired to the…
In this paper, we deal with the problem of synthesizing static output feedback controllers for stabilizing polynomial systems. Our approach jointly synthesizes a Lyapunov function and a static output feedback controller that stabilizes the…