Numerical method for a class of optimal control problems subject to nonsmooth functional constraints

AbstractIn this paper, we consider a class of optimal control problems which is governed by nonsmooth functional inequality constraints involving convolution. First, we transform it into an equivalent optimal control problem with smooth functional inequality constraints at the expense of doubling the dimension of the control variables. Then, using the Chebyshev polynomial approximation of the control variables, we obtain an semi-infinite quadratic programming problem. At last, we use the dual parametrization technique to solve the problem

Transformations for non-ideal uniform circular arrays operating in correlated signal environments

The Davies transformation is a method to transform the steering vector of a uniform circular array (UCA) to one with Vandermonde form. As such, it allows techniques such as spatial smoothing for direction-of-arrival (DOA) estimation in a correlated signal environment, developed originally for uniform linear arrays, to be applied to UCAs. However, the Davies transformation can be highly sensitive to perturbations of the underlying array model. This paper presents a method for deriving a more robust transformation using optimization techniques. The effectiveness of the method is illustrated through a number of DOA estimation examples

A study of optimization and optimal control computation : exact penalty function approach

In this thesis, We propose new computational algorithms and methods for solving four classes of constrained optimization and optimal control problems. In Chapter 1, we present a brief review on optimization and optimal control. In Chapter 2, we consider a class of continuous inequality constrained optimization problems. The continuous inequality constraints are first approximated by smooth function in integral form. Then, we construct a new exact penalty function, where the summation of all these approximate smooth functions in integral form, called the constraint violation, is appended to the objective function. In this way, we obtain a sequence of approximate unconstrained optimization problems. It is shown that if the value of the penalty parameter is sufficiently large, then any local minimizer of the corresponding unconstrained optimization problem is a local minimizer of the original problem. For illustration, three examples are solved using the proposed method.From the solutions obtained, we observe that the values of their objective functions are amongst the smallest when compared with those obtained by other existing methods available in the literature. More importantly, our method finds solutions which satisfy the continuous inequality constraints.In Chapter 3, we consider a general class of nonlinear mixed discrete programming problems. By introducing continuous variables to replace the discrete variables, the problem is first transformed into an equivalent nonlinear continuous optimization problem subject to original constraints and additional linear and quadratic constraints. However, the existing gradient-based optimization techniques have difficulty to solve this equivalent nonlinear optimization problem effectively due to the new quadratic inequality constraint. Thus, an exact penalty function is employed to construct a sequence of unconstrained optimization problems, each of which can be solved effectively by unconstrained optimization techniques, such as conjugate gradient or quasi-Newton types of methods.It is shown that any local optimal solution of the unconstrained optimization problem is a local optimal solution of the transformed nonlinear constrained continuous optimization problem when the penalty parameter is sufficiently large. Numerical experiments are carried out to test the efficiency of the proposed method.In Chapter 4, we investigate the optimal design of allpass variable fractional delay (VFD) filters with coefficients expressed as sums of signed powers-of-two terms, where the weighted integral squared error is minimized. A new optimization procedure is proposed to generate a reduced discrete search region. Then, a new exact penalty function method is developed to solve the optimal design of allpass variable fractional delay filter with signed powers-of-two coefficients. Design examples show that the proposed method is highly effective. Compared with the conventional quantization method, the solutions obtained by our method are of much higher accuracy. Furthermore, the computational complexity is low.In Chapter 5, we consider an optimal control problem in which the control takes values from a discrete set and the state and control are subject to continuous inequality constraints. By introducing auxiliary controls and applying a time-scaling transformation, we transform this optimal control problem into an equivalent optimal control problem subject to original constraints and additional linear and quadratic constraints, where the decision variables are taking values from a feasible region, which is the union of some continuous sets. However, due to the new quadratic constraints, standard optimization techniques do not perform well when they are applied to solve the transformed problem directly.We introduce a novel exact penalty function to penalize constraint violations, and then append this penalty function to the objective function, forming a penalized objective function. This leads to a sequence of approximate optimal control problems, each of which can be solved by using optimal control techniques, and consequently, many optimal control software packages, such as MISER 3.4, can be used. Convergence results how that when the penalty parameter is sufficiently large, any local solution of the approximate problem is also a local solution of the original problem. We conclude this chapter with some numerical results for two train control problems.In Chapter 6, some concluding remarks and suggestions for future research directions are made

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

New development of the inclusive-cone-based method for linear optimization

The purpose of this dissertation is to present a simple method for linear optimization including linear programming and linear semi-infinite programming, which is termed “the inclusive-cone-based method”. Using the inclusive cone as an analytic tool, theoretical aspects of linear programming are investigated. Sensitivity analysis in linear programming is examined from the perspective of an inclusive cone. The relationship of inclusiveness between correlated linear programming problems is also studied. New inclusive-cone-based ladder algorithms are proposed to solve linear programming problems in inequality form. Numerical experiments are implemented to show effectiveness and efficiency of the new linear programming ladder algorithms. To start the ladder method for linear programming problems, a single artificial constraint technique is introduced to find an initial ladder. Further, in the context of a new category of linear programming problems, an inclusive-cone-based solvability criterion is established to distinguish that a linear programming problem is inclusive-feasible (i.e., optimal), noninclusive-feasible (i.e., unbounded), inclusive-infeasible or noninclusive-infeasible. The inclusive-cone-based method for linear programming is also generalized to linear semi-infinite programming. An optimality result, based upon the concept of the generalized base point, is established. With this optimality result as a theoretical foundation, a ladder algorithm for solving linear semi-infinite programming problems is developed. The new algorithm has several features: at each iteration it only deals with a small fraction of constraints; at each iteration it selects a constraint most violated along a “parameterized centreline”, by solving a one-dimensional global optimization problem using the efficient bridging algorithm; at each iteration the selection of the incoming constraint has a great degree of freedom, which is controlled by a parameter arising in the global optimization problem; it can detect infeasibility and unboundedness after a finite number of iterations; it obviates extra work for feasibility verification as it handles feasibility and optimality simultaneously. A simple convergent result is presented. Numerical behaviour of the algorithm is examined on several test problems