Using Semidefinite Programming to Minimize Polynomials

نویسنده

  • Ruchira S. Datta
چکیده

Optimization problems arise in widely varying contexts. The general optimization problem is to find the minimum value of a certain function, the objective, on a certain set, defined by constraints. To make such problems amenable to analysis, further restrictions must be imposed on the kinds of objectives and constraints that may arise. A priori, it might seem useful to require them to be polynomial. After all, the entire toolbox of algebra and calculus is available to deal with polynomials. They can be represented, evaluated, and manipulated easily, and they are often used to approximate more complicated functions. Partly for these reasons, they arise in many kinds of applications. As this course has demonstrated, it so happens that this is not the most fruitful restriction for optimization problems. Even problems with quadratic objectives and quadratic constraints may be difficult to solve. Rather, it is the field of convex optimization which has developed a body of theory and practice leading to computationally effective solution procedures. However, the aforementioned reasons why polynomial optimization would be desirable are still valid. Thus, this paper attempts to explain and demonstrate how to use the techniques of convex optimization to approximately (often, exactly) solve polynomial optimization problems. For concreteness, the problems will be posed as minimization problems. For simplicity, the constraints will be linear, or absent.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

A Recurrent Neural Network Model for Solving Linear Semidefinite Programming

In this paper we solve a wide rang of Semidefinite Programming (SDP) Problem by using Recurrent Neural Networks (RNNs). SDP is an important numerical tool for analysis and synthesis in systems and control theory. First we reformulate the problem to a linear programming problem, second we reformulate it to a first order system of ordinary differential equations. Then a recurrent neural network...

متن کامل

Cones of Hermitian matrices and trigonometric polynomials

In this chapter we study cones in the real Hilbert spaces of Hermitian matrices and real valued trigonometric polynomials. Based on an approach using such cones and their duals, we establish various extension results for positive semidefinite matrices and nonnegative trigonometric polynomials. In addition, we show the connection with semidefinite programming and include some numerical experiments.

متن کامل

Exact SDP relaxations for classes of nonlinear semidefinite programming problems

An exact semidefinite linear programming (SDP) relaxation of a nonlinear semidefinite programming problem is a highly desirable feature because a semidefinite linear programming problem can efficiently be solved. This paper addresses the basic issue of which nonlinear semidefinite programming problems possess exact SDP relaxations under a constraint qualification. We do this by establishing exa...

متن کامل

Semi-Infinite Programming using High-Degree Polynomial Interpolants and Semidefinite Programming

In a common formulation of semi-infinite programs, the infinite constraint set is a requirement that a function parametrized by the decision variables is nonnegative over an interval. If this function is sufficiently closely approximable by a polynomial or a rational function, then the semi-infinite program can be reformulated as an equivalent semidefinite program. Solving this semidefinite pro...

متن کامل

Semidefinite Programming and Sums of Hermitian Squares of Noncommutative Polynomials

An algorithm for finding sums of hermitian squares decompositions for polynomials in noncommuting variables is presented. The algorithm is based on the “Newton chip method”, a noncommutative analog of the classical Newton polytope method, and semidefinite programming.

متن کامل

Discrete Transforms, Semidefinite Programming, and Sum-of-Squares Representations of Nonnegative Polynomials

Abstract. We present a new semidefinite programming formulation of sum-of-squares representations of nonnegative polynomials, cosine polynomials and trigonometric polynomials of one variable. The parametrization is based on discrete transforms (specifically, the discrete Fourier, cosine and polynomial transforms) and has a simple structure that can be exploited by straightforward modifications ...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2001