Keywordssolution description, quadratic programming, positive semi definite, affine set. Pdf solving semiinfinite programming problems by using an. A exible convergence proof is provided to cover di erent. Its arithmetic convergence rate is proved by taking in. Regularity and stability in nonlinear semiinfinite optimization. Several globally convergent schemes for solving sip problems have been proposed 1, 2, 5, 10, 11, 12. Consider the following convex quadratic semiinfinite programming problem. An adaptive dual parametrization algorithm for quadratic semi. 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. In section iii, an improved dual parameterization algorithm for solving the semi infinite programming problem is proposed. A new method for the design of adaptive array processors based on semiinfinite quadratic optimisation techniques is presented. In this paper, we consider a class of linearquadratic semiinfinite programming problems. In this paper we combine a classical adaptive discretization method developed by blankenship and falk and techniques regarding a semiinfinite optimization. Semiinfinite programming rembert reemtsen springer.
The methods considered here are remeztype algorithms also called exchange methods, e. A dual parameterization algorithm is also proposed for numerical solution of such problems. The simple description of such problems comes at a price. 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. Several relaxation techniques and their combinations are proposed and discussed. 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. A flexible convergence proof is provided to cover different. The hessian can be obtained from the quadratic terms by. 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. Quadratic programming 4 example 14 solve the following problem.
The mathematical representation of the quadratic programming qp problem is maximize. In chapter 5 k is used to denote the number of sip iterations performed in solving a sip. Note that the semiinfinite constraints are onedimensional, that is, vectors. 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. Such an nlp is called a quadratic programming qp problem. A cutting plane algorithm for solving convex quadratic semi infinite programming problems is presented.
Sip is an exciting part of mathematical programming. The objective function of the semiinfinite programme. Semiinfinite programming can be used to model a large variety of complex optimization problems. On optimality conditions for generalized semiinfinite programming problems, journal of optimization theory and applications, vol. Feasible sequential quadratic programming for finely discretized. We consider an inverse problem raised from the semidefinite quadratic programming sdqp problem. As can be seen, the q matrix is positive definite so the kkt conditions are necessary and sufficient for a global optimum. 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. In the proposed method, to compute a search direction in the primal space, we inexactly solve a semiinfinite convex quadratic program approximating the siplog. This paper studies the cuttingplane approach for solving quadratic semi infinite programming problems. Consider the following convex quadratic semi infinite programming problem. Asemiinfinitequadratic a semiinfinite quadratic programming algorithm with applications to array pattern synthesis.
Introduction consider a general doublyinfinite quadratic programming problem of the following form. An interior point sequential quadratic programmingtype method for. This paper presents a new extended active set strategy for optimizing antenna arrays by semi infinite quadratic programming. Pdf solving quadratic semiinfinite programming problems.
Journal of computational and applied mathematics 129. We rely on an oracle, a subroutine that performs this selection process. 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. 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.
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. Because the constraints must be in the form k i x,w i. A cutting plane algorithm for solving convex quadratic semiinfinite programming problems is presented. The technique finds broad use in operations research and is occasionally of use in statistical work. Sourcecodedocument ebooks document windows develop internetsocketnetwork game program. In section iv, a numerical experiment for this nonuniform filter bank design problem is presented. Pdf a new interface between matlab and sipampl was created, allowing the. An adaptive discretization method solving semiinfinite. Semiin nite programming for trajectory optimization with. Because of its many applications, quadratic programming is often viewed as a discipline in and of itself. Keywordssolution description, quadratic programming, positive semidefinite, affine set. This paper studies the cuttingplane approach for solving quadratic semiinfinite programming problems.
S2 quadratic programming a linearly constrained optimization problem with a quadratic objective function is called a quadratic program qp. An interior point sequential quadratic programmingtype. A semiinfinite quadratic programming algorithm with. In this paper, we present and improved adaptive algorithm for quadratic sip problems with positive definite objective and multiple linear infinite constraints. Asemiinfinitequadratic a semiinfinite quadrati codebus. Introduction consider a general doubly infinite quadratic programming problem of the following form. An adaptive dual parametrization algorithm for quadratic. Global optimization algorithms for semiinfinite and generalized. Relaxations of the cutting plane method for quadratic semi. Optimal solution characterization for infinite positive. A proof, based on the duality theorem of linear programming, is given for a duality theorem for a class of quadratic programs. March 9, 2008 abstract in this work, we develop duality of the minimum volume circumscribed ellipsoid and the maximum volume inscribed ellipsoid problems.
Nonlinear nonconvex semiinfinite programming with norm. Optimal solution approximation for infinite positivedefinite. The optimality criterion for the equalizer is either to minimize the complex deviation in the passband or to minimize its. On optimality conditions for generalized semi infinite programming problems, journal of optimization theory and applications, vol. Pdf relaxations of the cutting plane method for quadratic. A method is presented for minimizing a definite quadratic function under an infinite number of linear inequality restrictions. The objective function of the semi infinite programme.
Prob lems of this type naturally arise in approximation theory, optimal. Optimal solution characterization for infinite positive semi. This paper presents a new extended active set strategy for optimizing antenna arrays by semiinfinite quadratic programming. The optimality criterion is either to maximize the directivity of the. A proof, based on the duality theorem of linear programming, is given. Let s be a compact metric space and consider the convex semiin. In finite nonlinear optimization, around a feasible point x. Timevarying systems, positivedefinite costs, infinite horizon optimization, infinite quadratic programming, solution approx imations, lq control problems. Solving quadratic semiinfinite programming problems by using.
Description of semiinfinite programming from informs institute for operations research and management science. A semiinfinite programming problem is an optimization problem in which finitely many variables. Although we have only addressed the convex quadratic semi infinite programming problems in this paper, the convergence proofs derived here can be used as the basis for designing relaxed cuttingplane methods for solving convex semi infinite 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. A semiinfinite quadratic programming algorithm with applications to channel equalization. A projected lagrangian algorithm for semiinfinite programming. 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. 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.
A semi infinite quadratic a semi infinite quadratic programming algorithm with applications to array pattern synthesis. 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 specified peak side lobe level. The method is to minimise array output power subject to linear functional inequality constraints. Pdf solving quadratic semiinfinite programming problems by. A sequential quadratic programming algorithm designed to efficiently solve nonlinear. We then apply this result to the analysis of a multitarget linearquadratic control problem on semiinfinite time interval. 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. More specifically, we propose an interior point sequential quadratic programmingtype method that inexactly solves a sequence of semiinfinite quadratic programs approximating the siplog. Conversely, many nonlinear convex constraints can be expressed as lmis see the. A new exchange method for convex semiinfinite programming. We consider an inverse problem raised from the semi definite quadratic programming sdqp problem. We formulate this problem as a minimization problem with a positive semidefinite cone constraint and its dual is. A new quadratic semiinfinite programming algorithm based.
Semiinfinite quadratic optimisation method for the design. Analyzing the effect of uncertainty using semiinfinite. This model naturally arises in an abundant number of applications in different. Introduction consider a general convex programming problem c of the following form. Chapter 483 quadratic programming introduction quadratic programming maximizes or minimizes a quadratic objective function subject to one or more constraints. The so called dual parameterization method for quadratic semi infinite programming sip problems is developed recently.
On convergence of augmented lagrangian method for inverse. Pdf a semiinfinite quadratic programming algorithm with. Reduction to a simple quadratic programming problem. Package quadprog november 20, 2019 type package title functions to solve quadratic programming problems version 1. This example shows how to use semiinfinite programming to investigate the effect of uncertainty in the model parameters of an optimization problem.
We will formulate and solve an optimization problem using the function fseminf, a semiinfinite programming solver in optimization toolbox. 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 nonlinear semiinfinite program that minimizes a function including a logdeterminant logdet function. Using the duality theory, the dual problem is obtained, where the decision variables are measures. In section iii, an improved dual parameterization algorithm for solving the semiinfinite programming problem is proposed. An illustrative application is made in the theory of elastic structures. This paper is based on dnca dual nested complex approximation for optimizing communication channel equalizers using semi infinite quadratic programming. 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. 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.
Sip problems include finitely many variables and, in contrast to finite optimization problems, infinitely many 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 is based on dnca dual nested complex approximation for optimizing communication channel equalizers using semiinfinite quadratic programming. Turlach r port by andreas weingessel fortran contributions from cleve moler dposllinpack and. 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. Specifically, we propose a sequential quadratic programming sqptype algorithm equipped with techniques from the interior point method for sdps. We show that the coefficients of the quadratic function admit a simple expressions in term of the original data. The main purpose of the paper is to develop an algorithm for computing a karushkuhntucker kkt point for the siplog efficiently.