Related papers: A symbolic algorithm for exact polynomial optimiza…
The main objective of this paper is to derive the optimality conditions for one type of fuzzy optimization problems. At the beginning, we define a cone of descent direction for fuzzy optimization, and prove that its intersection with the…
We consider a class of optimization problems that involve determining the maximum value that a function in a particular class can attain subject to a collection of difference constraints. We show that a particular linear programming…
An algorithm to generate a minimal comprehensive Gr\"obner\, basis of a parametric polynomial system from an arbitrary faithful comprehensive Gr\"obner\, system is presented. A basis of a parametric polynomial ideal is a comprehensive…
We present a new algorithm for solving a polynomial program P based on the recent "joint + marginal" approach of the first author for, parametric optimization. The idea is to first consider the variable x1 as a parameter and solve the…
We give a high precision polynomial-time approximation scheme for the supremum of any honest n-variate (n+2)-nomial with a constant term, allowing real exponents as well as real coefficients. Our complexity bounds count field operations and…
In this paper, we describe a new method to compute the minimum of a real polynomial function and the ideal defining the points which minimize this polynomial function, assuming that the minimizer ideal is zero-dimensional. Our method is a…
We consider the hardness of approximation of optimization problems from the point of view of definability. For many NP-hard optimization problems it is known that, unless P = NP, no polynomial-time algorithm can give an approximate solution…
This work aims to introduce the framework of polynomial optimization theory to solve fractional polynomial problems (FPPs). Unlike other widely used optimization frameworks, the proposed one applies to a larger class of FPPs, not…
Let $f, f_1, \ldots, f_\nV$ be polynomials with rational coefficients in the indeterminates $\bfX=X_1, \ldots, X_n$ of maximum degree $D$ and $V$ be the set of common complex solutions of $\F=(f_1,\ldots, f_\nV)$. We give an algorithm…
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,…
We consider an equation of multiple variables in which a partial derivative does not vanish at a point. The implicit function theorem provides a local existence and uniqueness of the function for the equation. In this paper, we propose an…
Complex polynomial optimization has recently gained more and more attention in both theory and practice. In this paper, we study the optimization of a real-valued general conjugate complex form over various popular constraint sets including…
We initiate a systematic study of utilizing predictions to improve over approximation guarantees of classic algorithms, without increasing the running time. We propose a systematic method for a wide class of optimization problems that ask…
In a widely-studied class of multi-parametric optimization problems, the objective value of each solution is an affine function of real-valued parameters. Then, the goal is to provide an optimal solution set, i.e., a set containing an…
We study optimization problems that are neither approximable in polynomial time (at least with a constant factor) nor fixed parameter tractable, under widely believed complexity assumptions. Specifically, we focus on Maximum Independent…
Polynomial system solving is a classical problem in mathematics with a wide range of applications. This makes its complexity a fundamental problem in computer science. Depending on the context, solving has different meanings. In order to…
Multi-homogeneous polynomial systems arise in many applications. We provide bit complexity estimates for solving them which, up to a few extra other factors, are quadratic in the number of solutions and linear in the height of the input…
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…
We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed. For the optimization of an…
When considering an unconstrained minimization problem, a standard approach is to solve the optimality system with a Newton method possibly preconditioned by, e.g., nonlinear elimination. In this contribution, we argue that nonlinear…