# Semi infinite quadratic programming pdf

Chapter 483 quadratic programming introduction quadratic programming maximizes or minimizes a quadratic objective function subject to one or more constraints. Feasible sequential quadratic programming for finely discretized. A semiinfinite quadratic programming algorithm with applications to channel equalization. As can be seen, the q matrix is positive definite so the kkt conditions are necessary and sufficient for a global optimum. A complete, free, open source semi infinite programming tutorial is available here from elsevier as a pdf download from their journal of computational and applied mathematics, volume 217, issue 2, 1 august 2008, pages 394419. Specifically, we propose a sequential quadratic programming sqptype algorithm equipped with techniques from the interior point method for sdps. A projected lagrangian algorithm for semiinfinite programming. A proof, based on the duality theorem of linear programming, is given for a duality theorem for a class of quadratic programs. A new exchange method for convex semiinfinite programming. Introduction consider a general convex programming problem c of the following form. Contents basic concepts algorithms online and software resources references back to continuous optimization basic concepts semi infinite programming sip problems are optimization problems in which there is an infinite number of variables or an infinite number of constraints but not both.

Because of its many applications, quadratic programming is often viewed as a discipline in and of itself. Global optimization algorithms for semiinfinite and generalized. More specifically, we propose an interior point sequential quadratic programmingtype method that inexactly solves a sequence of semiinfinite quadratic programs approximating the siplog. Consider the following convex quadratic semi infinite programming problem. We consider an inverse problem raised from the semi definite quadratic programming sdqp problem. An illustrative application is made in the theory of elastic structures. March 9, 2008 abstract in this work, we develop duality of the minimum volume circumscribed ellipsoid and the maximum volume inscribed ellipsoid problems. A semiinfinite quadratic programming algorithm with. In the inverse problem, the parameters in the objective function of a given sdqp problem are adjusted as little as possible so that a known feasible solution becomes the optimal one. Turlach r port by andreas weingessel fortran contributions from cleve moler dposllinpack and. The main purpose of the paper is to develop an algorithm for computing a karushkuhntucker kkt point for the siplog efficiently. Regularity and stability in nonlinear semiinfinite optimization. Nonlinear nonconvex semiinfinite programming with norm.

Semidefinite programming sdp is a subfield of convex optimization concerned with the optimization of a linear objective function a userspecified function that the user wants to minimize or maximize over the intersection of the cone of positive semidefinite matrices with an affine space, i. Several relaxation techniques and their combinations are proposed and discussed. Although we have only addressed the convex quadratic semiinfinite programming problems in this paper, the convergence proofs derived here can be used as the basis for designing relaxed cuttingplane methods for solving convex semiinfinite programming problems. In mathematics, a definite quadratic form is a quadratic form over some real vector space v that has the same sign always positive or always negative for every nonzero vector of v. In optimization theory, semiinfinite programming sip is an optimization problem with a finite number of variables and an infinite number of constraints, or an infinite number of variables and a. In this paper, we consider a class of linear quadratic semi infinite programming problems. Asemiinfinitequadratic a semiinfinite quadratic programming algorithm with applications to array pattern synthesis. Keywordssolution description, quadratic programming, positive semidefinite, affine set. Semiinfinite programming rembert reemtsen springer.

This paper studies the cuttingplane approach for solving quadratic semi infinite programming problems. An adaptive discretization method solving semiinfinite. Pdf relaxations of the cutting plane method for quadratic. Consider the following convex quadratic semiinfinite programming problem. The method is to minimise array output power subject to linear functional inequality constraints. According to that sign, the quadratic form is called positivedefinite or negativedefinite a semidefinite or semidefinite quadratic form is defined in much the same way, except that always positive and. This paper studies the cuttingplane approach for solving quadratic semiinfinite programming problems. A cutting plane algorithm for solving convex quadratic semiinfinite programming problems is presented. Semiin nite programming 5 constraint generation oracle the key question for constraint generation is which new constraint g ik x. Pdf a semiinfinite quadratic programming algorithm with. Sip problems include finitely many variables and, in contrast to finite optimization problems, infinitely many inequality constraints.

Quadratic programming 4 example 14 solve the following problem. Relaxations of the cutting plane method for quadratic semi. Price 7 1 introduction 7 2 exact penalty funetions for semi infinite programming 143 3 trust region versus line search algorithms 145 4 the multilocal optimization subproblem 148. Note that the semiinfinite constraints are onedimensional, that is, vectors. In finite nonlinear optimization, around a feasible point x. We formulate this problem as a minimization problem with a positive semidefinite cone constraint and its dual is. Let s be a compact metric space and consider the convex semiin. Using the duality theory, the dual problem is obtained, where the decision variables are measures. We will formulate and solve an optimization problem using the function fseminf, a semiinfinite programming solver in optimization toolbox.

Optimal solution characterization for infinite positive. More specifically, we propose an interior point sequential quadratic programming type method that inexactly solves a sequence of semi infinite quadratic programs approximating the siplog. Asemiinfinitequadratic a semiinfinite quadrati codebus. A exible convergence proof is provided to cover di erent.

In section iii, an improved dual parameterization algorithm for solving the semi infinite programming problem is proposed. Semiin nite programming for trajectory optimization with. Timevarying systems, positivedefinite costs, infinite horizon optimization, infinite quadratic programming, solution approx imations, lq control problems. The optimality criterion is either to maximize the directivity of the. The objective function of the semi infinite programme. Semi in nite programming 5 constraint generation oracle the key question for constraint generation is which new constraint g ik x. In this paper, we consider a class of linearquadratic semiinfinite programming problems. Price 7 1 introduction 7 2 exact penalty funetions for semiinfinite programming 143 3 trust region versus line search algorithms 145 4 the multilocal optimization subproblem 148. Prob lems of this type naturally arise in approximation theory, optimal.

This example shows how to use semiinfinite programming to investigate the effect of uncertainty in the model parameters of an optimization problem. An adaptive dual parametrization algorithm for quadratic semi. Sourcecodedocument ebooks document windows develop internetsocketnetwork game program. Semiinfinite programming, that allows for either infinitely many constraints or infinitely many variables but not both, is a natural extension of ordinary mathematical programming. This paper presents a new extended active set strategy for optimizing antenna arrays by semiinfinite quadratic programming. Analyzing the effect of uncertainty using semiinfinite. A method is presented for minimizing a definite quadratic function under an infinite number of linear inequality restrictions. Semiinfinite programming can be used to model a large variety of complex optimization problems.

A sequential quadratic programming algorithm designed to efficiently solve nonlinear. Contents basic concepts algorithms online and software resources references back to continuous optimization basic concepts semiinfinite programming sip problems are optimization problems in which there is an infinite number of variables or an infinite number of constraints but not both. Conversely, many nonlinear convex constraints can be expressed as lmis see the. The optimality criterion for the equalizer is either to minimize the complex deviation in the passband or to minimize its stopband energy when subjected to a. An adaptive dual parametrization algorithm for quadratic. The so called dual parameterization method for quadratic semi infinite programming sip problems is developed recently. We consider an inverse problem raised from the semidefinite quadratic programming sdqp problem. A new method for the design of adaptive array processors based on semiinfinite quadratic optimisation techniques is presented.

A semi infinite quadratic a semi infinite quadratic programming algorithm with applications to array pattern synthesis. Stanford university, 1989 abstract semiinfinite programming, that allows for either infinitely many constraints or infinitely many variables but not both, is a natural extension of ordinary mathematical programming. Abstract this paper studies the cutting plane method for solving quadratic semiin nite programming problems. Special features of the method are that it generates a sequence of feasible solutions and a sequence of basic solutions simultaneously and that it has very favourable properties concerning numerical stability. The optimality criterion for the equalizer is either to minimize the complex deviation in the passband or to minimize its. Optimal solution characterization for infinite positive semi. A cutting plane algorithm for solving convex quadratic semi infinite programming problems is presented. The simple description of such problems comes at a price. The methods considered here are remeztype algorithms also called exchange methods, e.

In section iii, an improved dual parameterization algorithm for solving the semiinfinite programming problem is proposed. In the proposed method, to compute a search direction in the primal space, we inexactly solve a semiinfinite convex quadratic program approximating the siplog. Solving quadratic semiinfinite programming problems by using. In this paper, we present and improved adaptive algorithm for quadratic sip problems with positive definite objective and multiple linear infinite constraints. This model naturally arises in an abundant number of applications in different. An interior point sequential quadratic programmingtype. Lewis partially finite convex programming i 17 with c a cone we obtain duality results for semiinfinite linear programming and variants, and for certain quadratic programs in the hilbert space of square integrable functions, l2t, p. The technique finds broad use in operations research and is occasionally of use in statistical work. Sip is an exciting part of mathematical programming. The objective function of the semiinfinite programme. Keywordssolution description, quadratic programming, positive semi definite, affine set. Various errors and mismatches between ideal cases and actual scenario are incorporated into the constraints, which set the allowable upper and lower bounds on the array. On convergence of augmented lagrangian method for inverse. Pdf solving quadratic semiinfinite programming problems by.

A new quadratic semiinfinite programming algorithm based. This paper is based on dnca dual nested complex approximation for optimizing communication channel equalizers using semiinfinite quadratic programming. Optimal solution approximation for infinite positive. Solving quadratic semiinfinite programming problems by. A flexible convergence proof is provided to cover different. Several globally convergent schemes for solving sip problems have been proposed 1, 2, 5, 10, 11, 12.

Lewis partially finite convex programming i 17 with c a cone we obtain duality results for semi infinite linear programming and variants, and for certain quadratic programs in the hilbert space of square integrable functions, l2t, p. In this paper, we consider a nonlinear semiinfinite program that minimizes a function including a logdeterminant logdet function. The hessian can be obtained from the quadratic terms by. A semiinfinite programming problem is an optimization problem in which finitely many variables. Its arithmetic convergence rate is proved by taking in. Because the constraints must be in the form k i x,w i. In section iv, a numerical experiment for this nonuniform filter bank design problem is presented. Pdf solving quadratic semiinfinite programming problems. S2 quadratic programming a linearly constrained optimization problem with a quadratic objective function is called a quadratic program qp. A dual parameterization algorithm is also proposed for numerical solution of such problems. Journal of computational and applied mathematics 129. Reduction to a simple quadratic programming problem. Abstract this paper studies the cutting plane method for solving quadratic semi in nite programming problems.

This paper is based on dnca dual nested complex approximation for optimizing communication channel equalizers using semi infinite quadratic programming. Pdf a new interface between matlab and sipampl was created, allowing the. Package quadprog november 20, 2019 type package title functions to solve quadratic programming problems version 1. Pdf solving semiinfinite programming problems by using an. We rely on an oracle, a subroutine that performs this selection process.

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