On Solving Large-Scale Finite Minimax Problems using Exponential Smoothing

Journal of Optimization Theory and Applications, Vol. 148, No. 2, pp. 390-421

A Computationally Efficient Feasible Sequential Quadratic Programming Algorithm

The role of optimization in both engineering analysis and designis continually expanding. As such, faster and more powerful optimization algorithms are in constant demand. In this dissertation, motivated by problems from engineering analysis and design, new Sequential Quadratic Programming (SQP) algorithms generating feasible iterates are described and analyzed. What distinguishes these algorithms from previous feasible SQP algorithms is a dramatic reduction in the amount of computation required to generate a new iterate while still enjoying the same global and fast local convergence properties.First, a basic algorithm which solves the standard smooth inequality constrained nonlinear programming problem is considered. The main idea involves a simple perturbation of the Quadratic Program (QP) for the standard SQP search direction. The perturbation has the property that a feasible direction is always obtained and fast local convergence is preserved. An extension of the basic algorithm is then proposed which solves the inequality constrained mini-max problem. The algorithm exploits the special structure of the problem and is shown to have the same global and local convergence properties as the basic algorithm.Next, the algorithm is extended to efficiently solve problems with very many objective and/or constraint functions. Such problems often arise in engineering design as, e.g., discretized Semi-Infinite Programming (SIP) problems. The key feature of the extension is that only a small subset of the objectives and constraints are used to generate a search directionat each iteration. The result is much smaller QP sub-problems and fewer gradient evaluations.The algorithms all have been implemented and tested. Preliminary numericalresults are very promising. The number of iterations and function evaluations required to converge to a solution are, on average, roughly the same as for a widely available state-of-the-art feasible SQP implementation, whereas the amount of computation required per iteration is much less. The ability of the algorithms to effectively solve real problems from engineering design is demonstrated by considering signal set design problems for optimal detection in the presence of non-Gaussian noise

A Mesh Reduction Approach to Parametric Surface Polygonization

Surface polygonization is the process by which a representative polygonal mesh of a surface is constructed for rendering or analysis purposes. This work presents a new surface polygonization algorithm specifically tailored to be applied to a large class of models which are created with parametric surfaces. This method has particular application in the area of building virtual environments from computer-aided-design (CAD) models. The algorithm is based on an edge reduction scheme that collapses two vertices of a given polygon edge onto one new vertex. A two step approach is implemented consisting of boundary edge reduction followed by interior edge reduction. A maximin optimization is used to determine the location of the new vertex. The concept of a visible region as the location space of the new vertex is introduced. The method presented here differs from existing methods in that it takes advantage of the fact that for many models, the exact surface representation of the model is known before the polygonization is attempted. Because the precise surface definition is known, a maximin optimization procedure, that uses the surface information, can be used to locate the new vertex. The algorithm attempts to overcome the deficiencies in existing techniques while minimizing the number of polygons required to represent a surface and still maintaining surface integrity in the rendered model. This paper presents the algorithm and testing results

An SQP Algorithm For Finely Discretized Continuous Minimax Problems And Other Minimax Problems With Many Objective Functions

. A common strategy for achieving global convergence in the solution of semi-infinite programming (SIP) problems, and in particular of continuous minimax problems, is to (approximately) solve a sequence of discretized problems, with a progressively finer discretization meshes. Finely discretized minimax and SIP problems, as well as other problems with many more objectives /constraints than variables, call for algorithms in which successive search directions are computed based on a small but significant subset of the objectives/constraints, with ensuing reduced computing cost per iteration and decreased risk of numerical difficulties. In this paper, an SQP-type algorithm is proposed that incorporates this idea in the particular case of minimax problems. The general case will be considered in a separate paper. The quadratic programming subproblem that yields the search direction involves only a small subset of the objective functions. This subset is updated at each iteration in such a wa..

An SQP Algorithm for Finely Discretized Continuous Minimax Problems and Other Minimax Problems with Many Objective Functions

A common strategy for achieving global convergence in the solution of semi-infinite programming (SIP) problems, and in particular of continuous minimax problems, is to (approximately) solve a sequence of discretized problems, with a progressively finer discretization mesh. Finely discretized minimax and SIP problems, as well as other problems with many more objectives/constraints than variables, call for algorithms in which successive search directions are computed based on a small but significant subset of the objectives/constraints, with ensuing reduced computing cost per iteration and decreased risk of numerical difficulties. In this paper, an SQP-type algorithm is proposed that incorporates this idea in the particular case of minimax problems. The general case will be considered in a separate paper. The quadratic programming subproblem that yields the search direction involves only a small subset of the objectives functions. This subset is updated at each iteration in such a way that global convergence is insured. Heuristics are suggested that take advantage of a possible close relationship between ﲡdjacent objective functions. Numerical results demonstrate the efficiency of the proposed algorithm

Um método de redução para programação semi-infinita não linear baseado numa técnica de penalidade exacta

Tese de doutoramento em Engenharia Industrial e de SistemasOs problemas de programação semi-infinita (PSI) aparecem nas mais diversas áreas da Engenharia, tais como, no planeamento da trajectória de robôs, no controlo da poluição atmosférica, no planeamento da produção, no desenho óptimo de conjuntos de sinais e desenhos de filtros digitais. Esta tese é dedicada a problemas de PSI não linear na sua forma mais geral. Os problemas considerados são caracterizados por possuírem um número finito de variáveis e um conjunto infinito de restrições. Os métodos numéricos existentes para a resolução de problemas de PSI podem ser divididos em três classes principais: métodos de discretização, métodos das trocas e métodos de redução. Os métodos de redução são os que possuem melhores propriedades teóricas de convergência. São, também, os mais exigentes em termos numéricos uma vez que exigem a resolução de problemas auxiliares, em que se pretende a determinação de todos os óptimos globais e locais (optimização multi-local). Nas últimas décadas foram apresentados vários algoritmos para problemas de PSI. Contudo há pouco software disponível e nenhum fornece uma implementação de um método pertencente à classe de redução. Neste trabalho é proposto um algoritmo de redução local baseado na técnica de penalidade. A função usada considera uma extensão de uma função de penalidade de norma-1 aumentada. A escolha desta função de penalidade para propor a extensão às restrições finitas deve-se à obtenção de melhores resultados numéricos para um conjunto de problemas teste de PSI sem restrições finitas, em comparação com as funções de penalidade baseadas na norma 1, 2 e ∞ da violação das restrições. Fez-se o estudo das propriedades teóricas da função de penalidade estendida. É feita uma implementação do algoritmo de redução local proposto. O solver desenvolvido é designado por SIRedAl (Semi-Infinite Reduction Algorithm). Este solver foi implementado em MATLAB e é capaz de resolver problemas de PSI na forma mais geral com dimensão infinita máxima de 2. O código do solver usa dois algoritmos diferentes na minimização da função de penalidade e dois na resolução dos problemas multi-locais. O solver foi testado com 117 problemas teste da base de dados SIPAMPL e os resultados numéricos confirmaram a potencialidade do algoritmo proposto.Semi-infinite programming (SIP) problems arise in several engineering areas such as, for example, robotic trajectory planning, production planning, digital filter design and air pollution control. This thesis is devoted to SIP problems in the most general form. These problems are characterized to have a finite number of variables and an infinite set of constraints. The existing numerical methods for solving SIP problems can be divided into three major classes: discretization, exchange and reduction type methods. The reduction type methods are the ones with better theoretical properties, but they are also the most de- manding in computation terms, since they require to solve sub-problems to the local and global optimality (multi-local optimization). In last decades several algorithms were proposed for SIP, but there are not many pu- blicly available software and none provides an implementation of a method belonging to the reduction type class. In this work we propose a reduction type algorithm based on a penalty technique. The penalty function used is an extension of a penalty function of 1-norm, allowing the inclu- sion of finite constraints. In order to define the best penalty function, a numerical study of penalty functions based on the standard 1, 2 and ∞ norms are performed, considering test problems without finite constraints. A theoretical study of the extended penalty function is also performed. The proposed reduction algorithm is implemented in a solver coined as SIRedAl (Semi- Infinite Reduction Algorithm). The solver has been implemented in MATLAB and is capable of solving SIP problems in the most general form with a maximum of two infinite variables. The solver code uses two different algorithms in the minimization of the penalty function and also two different algorithms for solving the multi-local problems. The solver has been tested with 117 test problems from the database SIPAMPL and numerical results confirmed the algorithm potential.Instituto Superior de Engenharia do Port

Global optimization algorithms for semi-infinite and generalized semi-infinite programs

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2008.Includes bibliographical references (p. 235-249).The goals of this thesis are the development of global optimization algorithms for semi-infinite and generalized semi-infinite programs and the application of these algorithms to kinetic model reduction. The outstanding issue with semi-infinite programming (SIP) was a methodology that could provide a certificate of global optimality on finite termination for SIP with nonconvex functions participating. We have developed the first methodology that can generate guaranteed feasible points for SIP and provide e-global optimality on finite termination. The algorithm has been implemented in a branch-and-bound (B&B) framework and uses discretization coupled with convexification for the lower bounding problem and the interval constrained reformulation for the upper bounding problem. Within the framework of SIP we have also proposed a number of feasible-point methods that all rely on the same basic principle; the relaxation of the lower-level problem causes a restriction of the outer problem and vice versa. All these methodologies were tested using the Watson test set. It was concluded that the concave overestimation of the SIP constraint using McCormcick relaxations and a KKT treatment of the resulting expression is the most computationally expensive method but provides tighter bounds than the interval constrained reformulation or a concave overestimator of the SIP constraint followed by linearization. All methods can work very efficiently for small problems (1-3 parameters) but suffer from the drawback that in order to converge to the global solution value the parameter set needs to subdivided. Therefore, for problems with more than 4 parameters, intractable subproblems arise very high in the B&B tree and render global solution of the whole problem infeasible.(cont.) The second contribution of the thesis was the development of the first finite procedure that generates guaranteed feasible points and a certificate of e-global optimality for generalized semi-infinite programs (GSIP) with nonconvex functions participating. The algorithm employs interval extensions on the lower-level inequality constraints and then uses discretization and the interval constrained reformulation for the lower and upper bounding subproblems, respectively. We have demonstrated that our method can handle the irregular behavior of GSIP, such as the non-closedness of the feasible set, the existence of re-entrant corner points, the infimum not being attained and above all, problems with nonconvex functions participating. Finally, we have proposed an extensive test set consisting of both literature an original examples. Similar to the case of SIP, to guarantee e-convergence the parameter set needs to be subdivided and therefore, only small examples (1-3 parameters) can be handled in this framework in reasonable computational times (at present). The final contribution of the thesis was the development of techniques to provide optimal ranges of valid reduction between full and reduced kinetic models. First of all, we demonstrated that kinetic model reduction is a design centering problem and explored alternative optimization formulations such as SIP, GSIP and bilevel programming. Secondly, we showed that our SIP and GSIP techniques are probably not capable of handling large-scale systems, even if kinetic model reduction has a very special structure, because of the need for subdivision which leads to an explosion in the number of constraints. Finally, we propose alternative ways of estimating feasible regions of valid reduction using interval theory, critical points and line minimization.by Panayiotis Lemonidis.Ph.D

Robot Manipulators

Robot manipulators are developing more in the direction of industrial robots than of human workers. Recently, the applications of robot manipulators are spreading their focus, for example Da Vinci as a medical robot, ASIMO as a humanoid robot and so on. There are many research topics within the field of robot manipulators, e.g. motion planning, cooperation with a human, and fusion with external sensors like vision, haptic and force, etc. Moreover, these include both technical problems in the industry and theoretical problems in the academic fields. This book is a collection of papers presenting the latest research issues from around the world