An exact penalty function for semiinfinite programming. Rosinger, a hellytype theorem and semiinfinite programming,constructive approaches to mathematical models academic press, new york, 1979 pp. A semiinfinite programming problem is an optimization problem in which finitely many variables appear in infinitely many constraints. This function is a generalisation of the d exact penalty function for nonlinear programming see, e. A quasinewton interior point method for semi infinite programming.
Gpu robot motion planning using semiinfinite nonlinear. Sqp methods can be obtained from a local reduction of \p\ to a finite program, which is inspired in the implicit function theorem. To get started, install this package from github using the devtools package. Gpu robot motion planning using semi infinite nonlinear programming.
A semiinfinite programming based algorithm for determining t. Ieee transactions on parallel and dis tributed systems, institute of electrical and electronics engineers, 2016, 27 10, pp. Penalty and smoothing methods for convex semiinfinite. In mathematics, a semiinfinite programming sip problem is an optimization problem with a finite number of variables and an infinite number of constraints. Main applications sample problems transient eddy currents in a semiinfinite solid. The algorithm implements an exchange type method for sip where the lower bound problem is addressed by a branch and bound method.
Sipampl is an environment for coding semiinfinite programming sip. An implementation of a discretization method for semiinfinite programming. This book presents the state of the art in sip in a suggestive way, bringing the powerful sip tools close to the potential users in different scientific and technological fields. It highlights topological and structural aspects of general semi. Gsip is defined as generalized semiinfinite programming mathematics somewhat frequently. Solving semiinfinite programming problems by using an. This is an example of the transient eddy currents simulation, performed with. Hauser, in workshop on the algorithmic foundations of robotics wafr, 2018. Section 7 treats numerical methods based on either discretization or local reduction with the emphasis on the design of superlinearly convergent sqptype methods.
On duality theory of convex semiinfinite programming. One of the major computational tasks of using the traditional cutting plane approach to solve linear semiinfinite programming problems lies in finding a global optimizer of a nonlinear. Gpu robot motion planning using semi infinite nonlinear programming benjamin chretien, adrien escande, abderrahmane kheddar to cite this version. Relaxed cutting plane method for solving linear semiinfinite programming problems1 s. In this paper, a truncated projected newtontype algorithm is presented for solving largescale semiinfinite programming problems. Semiinfinite programming nonconvex optimization and its. This example shows how to use semiinfinite programming to investigate the effect of uncertainty in the model parameters of an optimization problem. Sequential quadratic programming sqp methods can be applied to problems that satisfy the same assumptions as those required for the kkt reduction methods. In a generalized semiinfinite programming gsip problem, the feasible set. This is a hybrid method of a truncated projected newton direction and a modified projected gradient direction. University of canterbury, christchurch, new zealand. Because the constraints must be in the form k i x,w i.
To mention some, among others, conic programming, semi definite programming, semi infinite programming and some meta heuristic techniques. Chapter 8 of semiinfinite programming proceedings of the international workshop on semiinfinite programming, r. Semiinfinite optimization with nonconvex geometries a package for optimizing robot poses and trajectories in the presence of nonconvex obstacle constraints using semiinfinite programming. In this paper we describe an exact penalty function for semiinfinite programming. The sipampl software package includes an interface to connect ampl to any sip solver, in particular to the nsips solver. Optimization methods and software volume 30, 2015 issue 3. How is generalized semiinfinite programming mathematics abbreviated.
Connections between semiinfinite and semidefinite programming. Several globally convergent schemes for solving sip problems have been proposed 1, 2, 5, 10, 11, 12. There, algebraic properties of finite linear programming are brought to bear on duality theory in semiinfinite programming. This is the first book that exploits the bilevel structure of semiinfinite programming systematically. A quasinewton interior point method for semiinfinite. The truncated projected newton method is used to solve the constrained nonlinear system. This model naturally arises in an abundant number of applications in different fields of mathematics, economics and engineering.
Rn, t is an infinite set, and all the functions are realvalued. Csip is freely available for educational, research, and commercial use, but we expect that all publications describing work using this software quote the reference given below. A thesis submitted in partial fulfillment of the requirements for the degree of doctor of philosophy in mathematics at the university of canterbury, by c. A semiinfinite solid is an idealized body that has a single plane surface and extends to infinity in all directions, as shown in fig. Relaxed cutting plane method for solving linear semi. Fiacco av, kortanek ko eds semiinfinite programming and applications. The key to a theoretical treatment of the topological features in the feasible set of gsip lies in the bilevel structure of semiinfinite. It aims to provide optimisation methods for sip problems.
We propose a method that computes a feasible point with an objective value better than or equal to a target value f0 or proves that such a point does not exist. Semiinfiniteoptimization kris hauser 10302018 kris. In this paper we consider minmax convex semiinfinite programming. The solver adresses a class of convex semiinfinite programming sip problems. Semiinfinite programming, that allows for either infinitely many constraints or infinitely many variables but not both, is a natural extension of ordinary mathematical 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. This package contains code accompanying the paper semiinfinite programming for trajectory optimization with nonconvex obstacles by k.
A semiinfinite programming algorithm for robot trajectory planning. Alaeddin malek, leila jafariankhaled abad, and samaneh khodayarisamghabadi. Analyzing the effect of uncertainty using semiinfinite. Global optimization of semiinfinite programs via restriction of the righthand side description this model provides an algorithm for the global solution of a semiinfinite program sip without convexity assumptions. Back to semiinfinite programming sequential quadratic programming sqp methods can be applied to problems that satisfy the same assumptions as those required for the kkt reduction methods. Semiinfinite programming, duality, discretization and optimality conditionsy alexander shapiro school of industrial and systems engineering, georgia institute of technology, atlanta, georgia 303320205, usa received 3 july 2008. Sip problems include finitely many variables and, in contrast to finite optimization problems, infinitely many inequality constraints. Semiinfinite programming sip deals with optimization problems in which either the number of decision variables or the number of constraints is finite. Some interesting semiinfinite optimization problems can be reduced to semidefinite optimization problems, and hence solved efficiently using recent interiorpoint methods. This framework subsumes wellknown classical algorithms, but also provides some new methods with interesting properties. Gsip stands for generalized semiinfinite programming mathematics.
A semiinfinite program sip is an optimization problem in finitely many variables x. It represents the cells to the right of the initial head position. Numerical results are shown with the discretization method, implemented in the nsips solver and it proved to be e. Direct theorems in semiinfinite convex programming. A bounding method for convex semiinfinite programming. Sip is an exciting part of mathematical programming. On duality theory of convex semiinfinite programming 539.
An adaptive numerical method for semiinfinite elliptic. A quasinewton interior point method for semiinfinite programming. A cutting surface algorithm for semiinfinite convex. Sip,semi infinite programming,semi infinite programs,semi infinite program,semi infinite programmings this article presents a short introduction to semiinfinite programming sip, which over the last two decades has become a vivid research area in mathematical. Benjamin chretien, adrien escande, abderrahmane kheddar. To solve these problems we introduce a unified framework concerning remeztype algorithms and integral methods coupled with penalty and smoothing methods. Sip problems include finitely many variables and, in contrast to finite optimization problems, infinitely many. Semiinfinite programming using highdegree polynomial. 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. It represents the cells to the left of the initial head position in reverse order. A turing machine with a semiinfinite tape has a left end but no right end. Many cases or problems in everyday life that involve optimization to solve them.
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. We will formulate and solve an optimization problem using the function fseminf, a semiinfinite programming solver in optimization toolbox. Semiinfinite programming recent advances miguel angel. It accompanies the paper semiinfinite programming for trajectory optimization with nonconvex obstacles by k. Semi infinite programming nsips three different methods discretization, interior point, sqp dual, uses authors special ampl interface c source and winbinaries. This idealized body is used to indicate that the temperature change in the part of the body in which we are interested the region close to the surface is due to the thermal conditions on a single surface. Zanotto, optimal trajectory planningfor industrial robots, advances in engineering software, 414,2010, 548556. Semiinfinite programming nsips three different methods discretization, interior point, sqp dual, uses authors special ampl interface c source and winbinaries. In the former case the constraints are typically parameterized. Discretization methods for semiinfinite programming do not provide a feasible point in a finite number of iterations. Corm and pietrzykowski 1977 and may be used as a merit function for semiinfinite programming methods. 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 finite number of constraints.
Lately much developed especially in the emergence of new techniques to solve the problem of optimization. Conversely, suppose that solp is nonempty and bounded. Then a binary search on the space of objective values can be performed to obtain a feasible, eoptimal solution. Note that the semiinfinite constraints are onedimensional, that is, vectors. Gsip generalized semiinfinite programming mathematics. Even the local nonclosedness phenomenon, which does not have any analog in finite or standard semiinfinite programming, is a stable phenomenon in gsip. If this function is sufficiently closely approximable by a polynomial or a rational function, then the semiinfinite program can be reformulated as. A truncated projected newtontype algorithm for large. Semiinfinite programming rembert reemtsen springer.
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