A Pattern Search Filter Method for Nonlinear Programming without Derivatives

Abstract

This paper formulates and analyzes a pattern search method for general constrained optimization based on filter methods for step acceptance. Roughly, a filter method accepts a step that improves either the objective function value or the value of some function that measures the constraint violation. The new algorithm does not compute or approximate any derivatives, penalty constants, or Lagrange multipliers. A key feature of the new algorithm is that it preserves the division into search and local poll steps, which allows the explicit use of inexpensive surrogates or random search heuristics in the search step. It is shown here that the algorithm identifies limit points at which optimality conditions depend on local smoothness of the functions and, to a greater extent, on the choice of a certain set of directions. Stronger optimality conditions are guaranteed for smoother functions and, in the constrained case, for a fortunate choice of the directions on which the algorithm depends. These directional conditions generalize those given previously for linear constraints, but they do not require a feasible starting point. In the absence of general constraints, the proposed algorithm and its convergence analysis generalize previous work on unconstrained, bound constrained, and linearly constrained generalized pattern search. The algorithm is illustrated on some test examples and on an industrial wing planform engineering design application.

Characteristic Shape Sequences for Measures on Images

“…Robustness of the solution to noise and experimental uncertainty is enforced by allowing the ellipses to contain a specified small percentage of pixels that lie outside the specified image level set. The optimization problem is solved numerically by applying a mesh adaptive direct search (MADS) algorithm [6] with a filter [5]. The filter allows intermediate solutions that violate constraints in order to provide a more robust global search of the parameter space.…”

“…Audet and Dennis [5] introduced a GPS filter method (filter-GPS), in which new iterates are generated whenever simple decrease in the objective or an aggregate constraint violation function is achieved. Filter methods were first introduced by Fletcher and Leyffer [12] as a way to globalize sequential linear programming (SLP) and sequential quadratic programming (SQP) without using a penalty function.…”

“…Filter methods were first introduced by Fletcher and Leyffer [12] as a way to globalize sequential linear programming (SLP) and sequential quadratic programming (SQP) without using a penalty function. The Audet-Dennis GPS filter approach avoids the pitfalls of penalty parameters, but its convergence to standard first-order stationary points is not guaranteed [5] due to GPS limitations.…”

“…The strategy for picking such points is left entirely to the user. Common choices are discussed in [4] or [5], for example.…”

“…Also consistent with [5], we define a second constraint violation function, h X = h + ψ X , where the indicator function ψ X is defined exactly as before. Thus, for x / ∈ X, h X (x) = ∞, and the function h is not evaluated.…”