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