Related papers: A Certificate of Unboundedness for Polynomial Opti…
We present precise bit and degree estimates for the optimal value of the polynomial optimization problem $f^*:=\text{inf}_{x\in \mathscr{X}}~f(x)$, where $\mathscr{X}$ is a semi-algebraic set satisfying some non-degeneracy conditions. Our…
We give an improved polynomial bound on the complexity of the equation solvability problem, or more generally, of finding the value sets of polynomials over finite nilpotent rings. Our proof depends on a result in additive combinatorics,…
Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…
We propose a black-box approach to reducing large semidefinite programs to a set of smaller semidefinite programs by projecting to random linear subspaces. We evaluate our method on a set of polynomial optimization problems, demonstrating…
This paper focuses on the study of a mathematical program with equilibrium constraints, where the objective and the constraint functions are all polynomials. We present a method for finding its global minimizers and global minimum using a…
Polynomial optimization problems often arise in sequences indexed by dimension, and it is of interest to compute bounds on the optimal values of all problems in the sequence. Examples include certifying inequalities between symmetric…
We introduce an algorithm which can be directly used to feasible and optimum search in linear programming. Starting from an initial point the algorithm iteratively moves a point in a direction to resolve the violated constraints. At the…
Polynomial approximations of functions are widely used in scientific computing. In certain applications, it is often desired to require the polynomial approximation to be non-negative (resp. non-positive), or bounded within a given range,…
Polynomial optimization problems represent a wide class of optimization problems, with a large number of real-world applications. Current approaches for polynomial optimization, such as the sum of squares (SOS) method, rely on large-scale…
In safety-critical applications that rely on the solution of an optimization problem, the certification of the optimization algorithm is of vital importance. Certification and suboptimality results are available for a wide range of…
Bayesian optimization methods have been successfully applied to black box optimization problems that are expensive to evaluate. In this paper, we adapt the so-called super effcient global optimization algorithm to solve more accurately…
In this work we study preprocessing for tractable problems when part of the input is unknown or uncertain. This comes up naturally if, e.g., the load of some machines or the congestion of some roads is not known far enough in advance, or if…
This paper discusses the split feasibility problem with polynomials. The sets are semi-algebraic, defined by polynomial inequalities. They can be either convex or nonconvex, either feasible or infeasible. We give semidefinite relaxations…
We study the problem of detecting zeros of continuous functions that are known only up to an error bound, extending the earlier theoretical work with explicit algorithms and experiments with an implementation. More formally, the robustness…
Certificates of non-negativity such as Putinar's Positivstellensatz have been used to obtain powerful numerical techniques to solve polynomial optimization (PO) problems. Putinar's certificate uses sum-of-squares (sos) polynomials to…
In this paper, we present a branch and bound algorithm for extracting approximate solutions to Global Polynomial Optimization (GPO) problems with bounded feasible sets. The algorithm is based on a combination of SOS/Moment relaxations and…
We provide a formula for the lower bound in the form of $|F| \ge K$, in such a way that the decision version of unweighted non-bipartite matching can be solved in polynomial time. ~The parameter $K$ can vary from instance to instance. We…
Bayesian optimization has recently emerged as a popular and efficient tool for global optimization and hyperparameter tuning. Currently, the established Bayesian optimization practice requires a user-defined bounding box which is assumed to…
Discrete-time robust optimal control problems generally take a min-max structure over continuous variable spaces, which can be difficult to solve in practice. In this paper, we extend the class of such problems that can be solved through a…
We study first-order methods with preconditioning for solving structured nonlinear convex optimization problems. We propose a new family of preconditioners generated by symmetric polynomials. They provide first-order optimization methods…