E. Polak

On the mathematical foundations of nondifferentiable optimization in engineering design

It is shown by example that a large class of engineering design problems can be transcribed into the form of a canonical optimization problem with inequality constraints involving mar functions. Such problems are commonly referred to as semi-infinite optimization problems. The bulk of this paper is devoted to the development of a mathematical theory for the construction of first order nondifferentiable optimization algorithms, related to phase I - phase II methods of feasible directions, which solve these semi-infinite optimization problems. The applicability of the theory is illustrated with examples that are relevant to engineering design.

Reliability-based optimal structural design by the decoupling approach

Abstract A decoupling approach for solving optimal structural design problems involving reliability terms in the objective function, the constraint set or both is discussed and extended. The approach employs a reformulation of each problem, in which reliability terms are replaced by deterministic functions. The reformulated problems can be solved by existing semi-infinite optimization algorithms and computational reliability methods. It is shown that the reformulated problems produce solutions that are identical to those of the original problems when the limit-state functions defining the reliability problem are affine. For nonaffine limit-state functions, approximate solutions are obtained by solving series of reformulated problems. An important advantage of the approach is that the required reliability and optimization calculations are completely decoupled, thus allowing flexibility in the choice of the optimization algorithm and the reliability computation method.

An algorithm for optimization problems with functional inequality constraints

This paper presents an algorithm for minimizing an objective function subject to conventional inequality constraints as well as to inequality constraints of the functional type: \max_{\omega \in \Omega} \phi(z,\omega) \leq 0 , where Ω is a closed interval in R , and z \in R^{n} is the parameter vector to be optimized. The algorithm is motivated by a standard earthquake engineering problem and the problem of designing linear multivariable systems. The stability condition (Nyquist criterion) and disturbance suppression condition for such systems are easily expressed as a functional inequality constraint.

Nondifferentiable optimization algorithm for designing control systems having singular value inequalities

It has been known for some time that many control system design requirements can be expressed as differentiable inequalities. More recently, it has been shown that important structural properties such as robustness and low noise sensitivity can be expressed as nondifferentiable inequalities involving the singular values of a system or return difference transfer function matrix. This paper presents an optimization algorithm which permits all these constraints to be considered.

Moving horizon control of nonlinear systems with input saturation, disturbances and plant uncertainty

We present a moving horizon feedback system, based on constrained optimal control algorithms, for nonlinear plants with input saturation, disturbances and plant uncertainty. The system is a non-conventional sampled-data system: its sampling periods vary from sampling instant to sampling instant, and the control during the sampling time is not constant, but determined by the solution of an open loop optimal control problem. We show that the proposed moving horizon control system is robustly stable and is capable of attenuating L ∞ bounded disturbances

On the design of linear multivariable feedback systems via constrained nondifferentiable optimization in H/sup infinity / spaces

The design discussed is of linear, lumped, time-invariant, multivariable feedback systems, subject to various frequency and time-domain performance specifications. The approach is based on the use of stabilizing controller parametrizations which result in the formulation of feedback system design problems as convex, nondifferentiable optimization problems. These problems are solvable by recently developed nondifferentiable optimization algorithms for the constrained minimization of regular, uniformly locally Lipschitz continuous functions in R/sup N/. >

Semi-Infinite Optimization

We devote this chapter to optimality conditions and algorithms for solving semi-infinite optimization problems. In particular, we will consider semi-infinite min-max problems of the form.

On the Extension of Constrained Optimization Algorithms from Differentiable to Nondifferentiable Problems

This paper presents three general schemes for extending differentiable optimization algorithms to nondifferentiable problems. It is shown that the Armijo gradient method, phase-I–phase-II methods of feasible directions and exact penalty function methods have conceptual analogs for problems with locally Lipschitz functions and implementable analogs for problems with semismooth functions. The exact penalty method has required the development of a new optimality condition.

Control system design via semi-infinite optimization: A review

This paper presents a survey of the basic aspects involved in the design of linear multivariable control systems via semi-infinite optimization. Specific topics treated are a) database and simulation requirements, b) techniques for the transcription of design specifications into semi-infinite inequalities, and c) semi-infinite optimization algorithms for control system design.

An outer approximation approach to reliability-based optimal design of structures

We present a new formulation of the problem of minimizing the initial cost of a structure subject to a minimum reliability requirement, expressed in terms of the so-called design points of the first-order reliability theory, i.e., points on limit-state surfaces that are nearest to the origin in a transformed standard normal space, as well as other deterministic constraints. Our formulation makes it possible to use outer approximations algorithms for the solution of such optimal design problems, eliminating some of the major objections associated with treating them as bilevel optimization problems. A numerical example is presented that illustrates the reliability and efficiency of the algorithm.