Outer Approximation Algorithm Research Articles

A new problem in control system design (and circuit design)

An optimisation problem arising in the design of circuits and control systems is formulated, and an outer approximation algorithm for solving it is presented. The problem is characterized by constraints that must be satisfied for all functions in a given subset of a function space. It is shown that an outer-approximation-type algorithm increases the number of variables and constraints at each iteration. A practical algorithm will require a constraint dropping scheme.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An outer-approximation algorithm for a class of mixed-integer nonlinear programs

An outer-approximation algorithm for a class of mixed-integer nonlinear programs

Relaxation strategy for the structural optimization of process flow sheets

Abstract: This paper presents an Equality Relaxation variant to the Outer-Approximation algorithm for solving mixed-integer nonlinear programming (MINLP) problems that arise in structural optimization of process flowsheets. The propsed algorithm has the important capability of being able to explicitly handle nonlinear equations within MINLP formulations that have linear integer variables and linear/nonlinear continuous varibales. It is shown that through the explicit treatment of nonlinear equations, the proposed algorithm avoids computational difficulties (e.g. singularities, destruction of sparsity) that are experienced with algebraic or numerical elimination schemes.Also, theoretical properties of the Equality-Relaxation algorithm are discussed, and its performance is demonstrated with a planning problem and a flowsheet synthesis problem. Finally, a simple procedure for structural sensitivity analysis is presented.

Global optimization under Lipschitzian constraints

We will present a new method for finding the global minimum of a Lipschitzian function under Lipschitzian constraints. The method consists in converting the given problem into one of globally minimizing a concave function subject to a convex and a reverse convex constraints. The resulting algorithm is of the same complexity as the outer approximation algorithm for a concave minimization problem.

An outer-approximation algorithm for a class of mixed-integer nonlinear programs

An outer-approximation algorithm is presented for solving mixed-integer nonlinear programming problems of a particular class. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the proposed algorithm effectively exploits the structure of the problems, and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a mixed-integer linear master program. Convergence and optimality properties of the algorithm are presented, as well as a general discussion on its implementation. Numerical results are reported for several example problems to illustrate the potential of the proposed algorithm for programs in the class addressed in this paper. Finally, a theoretical comparison with generalized Benders decomposition is presented on the lower bounds predicted by the relaxed master programs.

An outer-approximation algorithm for a class of mixed-integer nonlinear programs

An outer-approximation algorithm is presented for solving mixed-integer nonlinear programming problems of a particular class. Linearity of the integer (or discrete) variables, and convexity of the ...

Optimization Problems Arising in Control Systems Design

Control design is naturally formulated as an optimization problem in which the constraints may be conventional, non-differentiable (e.g. constraints on singular values for robustness or eigenvalues for stability) or infinite-dimensional (e.g. time domain constraints on step responses or frequency domain constraints for performance) and the design variables may be finite dimensional (e.g. parameters of a controller) or infinite dimensional (e.g. a non-linear control law). This paper concentrates on the latter type of problem, a typical problem being the simultaneous determination of a Lyapunov function and non-linear control law to stabilize a non-linear system. An outer approximation algorithm is presented and shown to produce converging subsequences. The algorithm is illustrated by its application to a simple non-linear control problem.

Optimization problems arising in control systems design

Control design is naturally formulated as an optimization problem in which the constraints may be conventional, non-differentiable (e.g. constraints on singular values for robustness or eigenvalues for stability) or infinite-dimensional (e.g. time domain constraints on step responses or frequency domain constraints for performance) and the design variables may be finite dimensional (e.g. parameters of a controller) or infinite dimensional (e.g. a non-linear control law). This paper concentrates on the latter type of problem, a typical problem being the simultaneous determination of a Lyapunov function and non-linear control law to stabilize a non-linear system. An outer approximation algorithm is presented and shown to produce converging subsequences. The algorithm is illustrated by its application to a simple non-linear control problem.

Outer approximation algorithm for nondifferentiable optimization problems

It is known that the problem of minimizing a convex functionf(x) over a compact subsetX of ℝ n can be expressed as minimizing max{g(x, y)|y ∈X}, whereg is a support function forf[f(x) ≥g(x, y), for ally ∈X andf(x)=g(x, x)]. Standard outer-approximation theory can then be employed to obtain outer-approximation algorithms with procedures for dropping previous cuts. It is shown here how this methodology can be extended to nonconvex nondifferentiable functions.

Outer approximation algorithm for nondifferentiable optimization problems

Outer approximation algorithm for nondifferentiable optimization problems

An Outer Approximation Algorithm for Solving General Convex Programs

A new outer approximation algorithm is proposed for solving general convex programs. A remarkable advantage of the algorithm over existing outer approximation methods is that the approximation of the constraint set is not cumulative. That is, the algorithm solves at each iteration a quadratic program whose constraints depend only on the current estimate of an optimal solution. Convergence of the algorithm is proved and possible applications are discussed.