We propose an algorithm which combines multi-directional search (MDS) with nonsmooth techniques such as bundling to solve several difficult synthesis problems in automatic control. Applications include static and fixe...
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We propose an algorithm which combines multi-directional search (MDS) with nonsmooth techniques such as bundling to solve several difficult synthesis problems in automatic control. Applications include static and fixed-order output feedback controller design, simultaneous stabilization, H2/H ∞ synthesis to cite just a few. We show in which way direct search techniques may be safeguarded by nonsmooth oracles in order to maintain convergence certificates in the presence of nonsmoothness. Our numerical testing includes numerous benchmark examples. For instance, our algorithm needs 0.41 seconds to compute a static output feedback stabilizing controller for the Boeing 767 flutter benchmark problem (E.J. Davison, 1990), a system with 55 states.
This paper contains a new convergence analysis for the Lewis and Torczon generalized patternsearch (GPS) class of methods for unconstrained and linearly constrained optimization. This analysis is motivated by a desir...
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This paper contains a new convergence analysis for the Lewis and Torczon generalized patternsearch (GPS) class of methods for unconstrained and linearly constrained optimization. This analysis is motivated by a desire to understand the successful behavior of the algorithm under hypotheses that are satisfied by many practical problems. Specifically, even if the objective function is discontinuous or extended-valued, the methods find a limit point with some minimizing properties. Simple examples show that the strength of the optimality conditions at a limit point depends not only on the algorithm, but also on the directions it uses and on the smoothness of the objective at the limit point in question. The contribution of this paper is to provide a simple convergence analysis that supplies detail about the relation of optimality conditions to objective smoothness properties and to the de ning directions for the algorithm, and it gives previous results as corollaries.
Many engineering optimization problems involve a special kind of discrete variable that can be represented by a number, but this representation has no significance. Such variables arise when a decision involves some s...
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Many engineering optimization problems involve a special kind of discrete variable that can be represented by a number, but this representation has no significance. Such variables arise when a decision involves some situation like a choice from an unordered list of categories. This has two implications: The standard approach of solving problems with continuous relaxations of discrete variables is not available, and the notion of local optimality must be defined through a user-specified set of neighboring points. We present a class of direct searchalgorithms to provide limit points that satisfy some appropriate necessary conditions for local optimality for such problems. We give a more expensive version of the algorithm that guarantees additional necessary optimality conditions. A small example illustrates the differences between the two versions. A real thermal insulation system design problem illustrates the efficacy of the user controls for this class of algorithms.
The Spherical Code (SC) problem has many important applications in such fields as physics, molecular biology, signal transmission, chemistry, engineering and mathematics. This paper presents a bilevel optimization for...
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The Spherical Code (SC) problem has many important applications in such fields as physics, molecular biology, signal transmission, chemistry, engineering and mathematics. This paper presents a bilevel optimization formulation of the SC problem. Based on this formulation, the concept of balanced spherical code is introduced and a new approach, the Point Balance algorithm (PBA), is presented to search for a 1-balanced spherical code. Since an optimal solution of the SC problem (an extremal spherical code) must be a 1-balanced spherical code, PBA can be applied easily to search for an extremal spherical code. In addition, given a certain criterion, PBA can generate efficiently an approximate optimal spherical code on a sphere in the n-dimensional space R-n. Some implementation issues of PBA are discussed and putative global optimal solutions of the Fekete problem in 3, 4 and 5-dimensional space are also reported. Finally, an open question about the geometry of Fekete points on the unit sphere in the 3-dimensional space is posed.
In the literature, thermal insulation systems with a fixed number of heat intercepts have been optimized with respect to intercept locations and temperatures. The number of intercepts and the types of insulators that ...
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In the literature, thermal insulation systems with a fixed number of heat intercepts have been optimized with respect to intercept locations and temperatures. The number of intercepts and the types of insulators that surround them were chosen by parametric studies. This was because the optimization methods used could not treat such categorical variables. Discrete optimization variables are categorical if the objective function or the constraints can not be evaluated unless the variables take one of a prescribed enumerable set of values. The key issue is that categorical variables can not be treated as ordinary discrete variables are treated by relaxing them to continuous variables with a side constraint that they be discrete at the solution. A new mixed variable programming (MVP) algorithm makes it possible to optimize directly with respect to mixtures of discrete, continuous, and categorical decision variables. The result of applying MVP is shown here to give a 65% reduction in the objective function over the previously published result for a thermal insulation model from the engineering literature. This reduction is largely because MVP optimizes simultaneously with respect to the number of heat intercepts and the choices from a list of insulator types as well as intercept locations and temperatures. The main purpose of this paper is to show that the mixed variable optimization algorithm can be applied effectively to a broad class of optimization problems in engineering that could not be easily solved with earlier methods.
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