Novel approach to accelerating Newton's method for sup-norm optimization arising inH ∞-control

We present algorithms for solving general sup-norm minimization problems over spaces of analytic functions, such as those arising inH control. We also give an analysis and some theory of these algorithms. Part of this is specific to analytic optimization, while part holds for general sup-norm optimization. In particular, we are proposing a type of Newton-type algorithm which actually uses very high-order terms. The novel feature is that higher-order terms can be chosen in many ways while still maintaining a second-order convergence rate. Then, a clever choice of higher-order terms greatly reduces computation time. Conceivably this technique can be modified to accelerate Newton algorithms in some other circumstances. Estimates of order of convergence as well as results of numerical tests are also presented.This work was partially supported by the Air Force Office of Scientific Research and the National Science Foundation.     

Generalized γ-Valid Cut Procedure for Concave Minimization

The concept of a -valid cutting plane has been used in many types of algorithms for solving concave minimization problems. Unfortunately, the procedures proposed to date for constructing these cuts are valid only under certain assumptions that often may not hold in practice. Chief among these is the requirement that the feasible region of the concave minimization problem in question have full dimension, and that the objective function of this problem be concave rather than quasiconcave. In this article, we propose, validate, and show how to implement a more general -valid cutting plane procedure which eliminates these restrictions.     

Solving a class of multiplicative programming problems via c-programming

In this note we show that many classes of global optimization problems can be treated most satisfactorily by classical optimization theory and conventional algorithms. We focus on the class of problems involving the minimization of the product of several convex functions on a convex set which was studied recently by Kunoet al. [3]. It is shown that these problems are typical composite concave programming problems and thus can be handled elegantly by c-programming [4]–[8] and its techniques.     

ℓ1-Minimization with Magnitude Constraints in the Frequency Domain

In this paper, we study the -optimal control problem with additional constraints on the magnitude of the closed-loop frequency response. In particular, we study the case of magnitude constraints at fixed frequency points (a finite number of such constraints can be used to approximate an -norm constraint). In previous work, we have shown that the primal-dual formulation for this problem has no duality gap and both primal and dual problems are equivalent to convex, possibly infinite-dimensional, optimization problems with LMI constraints. Here, we study the effect of approximating the convex magnitude constraints with a finite number of linear constraints and provide a bound on the accuracy of the approximation. The resulting problems are linear programs. In the one-block case, both primal and dual programs are semi-infinite dimensional. The optimal cost can be approximated, arbitrarily well from above and within any predefined accuracy from below, by the solutions of finite-dimensional linear programs. In the multiblock case, the approximate LP problem (as well as the exact LMI problem) is infinite-dimensional in both the variables and the constraints. We show that the standard finite-dimensional approximation method, based on approximating the dual linear programming problem by sequences of finite-support problems, may fail to converge to the optimal cost of the infinite-dimensional problem.     

The complementary convex structure in global optimization

We show the importance of exploiting the complementary convex structure for efficiently solving a wide class of specially structured nonconvex global optimization problems. Roughly speaking, a specific feature of these problems is that their nonconvex nucleus can be transformed into a complementary convex structure which can then be shifted to a subspace of much lower dimension than the original underlying space. This approach leads to quite efficient algorithms for many problems of practical interest, including linear and convex multiplicative programming problems, concave minimization problems with few nonlinear variables, bilevel linear optimization problems, etc...     

Properties and Solution Methods for Large Location—Allocation Problems

Location-allocation with l _p distances is studied. It is shown that this structure can be expressed as a concave minimization programming problem. Since concave minimization algorithms are not yet well developed, five solution methods are developed which utilize the special properties of the location-allocation problem. Using the rectilinear distance measure, two of these algorithms achieved optimal solutions in all 102 test problems for which solutions were known. The algorithms can be applied to much larger problems than any existing exact methods.     

H∞-Controllers for Time-Delay Systems Using Linear Matrix Inequalities

In this paper, H -control design is developed for nominally linear systems with input as well as state delays. Both stability and H -norm bound conditions are established for asymptotically stable controlled systems. Necessary and sufficient conditions for feedback control synthesis are established first by using two forms; the first has one term representing pure state feedback, and the second has two terms comprising pure state feedback plus delayed state feedback. Then, the corresponding synthesis conditions for the cases of static-output feedback and observer-based feedback controllers are developed. The results are cast conveniently into a linear matrix inequality (LMI) framework, which can be solved numerically by efficient interior-point methods. With the aid of the LMI control toolbox software, the theoretical work is illustrated by computer simulation of numerous examples.     

Parallel computing in nonconvex programming

In this paper, we are concerned with the development of parallel algorithms for solving some classes of nonconvex optimization problems. We present an introductory survey of parallel algorithms that have been used to solve structured problems (partially separable, and large-scale block structured problems), and algorithms based on parallel local searches for solving general nonconvex problems. Indefinite quadratic programming posynomial optimization, and the general global concave minimization problem can be solved using these approaches. In addition, for the minimum concave cost network flow problem, we are going to present new parallel search algorithms for large-scale problems. Computational results of an efficient implementation on a multi-transputer system will be presented.     

A branch-and-reduce approach to global optimization

This paper presents valid inequalities and range contraction techniques that can be used to reduce the size of the search space of global optimization problems. To demonstrate the algorithmic usefulness of these techniques, we incorporate them within the branch-and-bound framework. This results in a branch-and-reduce global optimization algorithm. A detailed discussion of the algorithm components and theoretical properties are provided. Specialized algorithms for polynomial and multiplicative programs are developed. Extensive computational results are presented for engineering design problems, standard global optimization test problems, univariate polynomial programs, linear multiplicative programs, mixed-integer nonlinear programs and concave quadratic programs. For the problems solved, the computer implementation of the proposed algorithm provides very accurate solutions in modest computational time.     

Convexification, Concavification and Monotonization in Global Optimization

We show in this paper that via certain convexification, concavification and monotonization schemes a nonconvex optimization problem over a simplex can be always converted into an equivalent better-structured nonconvex optimization problem, e.g., a concave optimization problem or a D.C. programming problem, thus facilitating the search of a global optimum by using the existing methods in concave minimization and D.C. programming. We first prove that a monotone optimization problem (with a monotone objective function and monotone constraints) can be transformed into a concave minimization problem over a convex set or a D.C. programming problem via pth power transformation. We then prove that a class of nonconvex minimization problems can be always reduced to a monotone optimization problem, thus a concave minimization problem or a D.C. programming problem.     

How to Extend the Concept of Convexity Cuts to Derive Deeper Cutting Planes

A new type of cutting plane, termed a decomposition cut, is introduced that can be constructed under the same assumptions as the well-known convexity cut. Therefore it can be applied in algorithms (e.g. cutting plane, branch-and-cut) for various problems of global optimization, such as concave minimization, bilinear programming, reverse-convex programming, and integer programming. In computational tests with cutting plane algorithms for concave minimization, decomposition cuts were shown to be superior to convexity cuts.     

Polyhedral annexaton,dualization and dimension reduction technique in global optimization

We demonstrate how the size of certain global optimization problems can substantially be reduced by using dualization and polyhedral annexation techniques. The results are applied to develop efficient algorithms for solving concave minimization problems with a low degree of nonlinearity. This class includes in particular nonconvex optimization problems involving products or quotients of affine functions in the objective function.This work was completed while the author was visiting the Department of Mathematics of Linköping University.     

Convergent Outer Approximation Algorithms for Solving Unary Programs

Interesting cutting plane approaches for solving certain difficult multiextremal global optimization problems can fail to converge. Examples include the concavity cut method for concave minimization and Ramana's recent outer approximation method for unary programs which are linear programming problems with an additional constraint requiring that an affine mapping becomes unary. For the latter problem class, new convergent outer approximation algorithms are proposed which are based on sufficiently deep l-norm or quadratic cuts. Implementable versions construct optimal simplicial inner approximations of Euclidean balls and of intersections of Euclidean balls with halfspaces, which are of general interest in computational convexity. Computational behavior of the algorithms depends crucially on the matrices involved in the unary condition. Potential applications to the global minimization of indefinite quadratic functions subject to indefinite quadratic constraints are shown to be practical only for very small problem sizes.     

Transfer Function Optimization Procedure for the H 2/H ∞ Problem

In this paper, the H ₂/H problem is considered in a transfer-function setting, i.e., without a priori chosen bounds on the controller order. An optimization procedure is described which is based on a parametrization of all feasible descending directions stemming from a given point of the feasible transfer-function set. A search direction at each such point can be obtained on the basis of the solution of a convex finite-dimensional problem which can be converted into a LMI problem. Moving along the chosen direction in each step, the procedure in question generates a sequence of feasible points whose cost functional values converge to the optimal value of the H ₂/H problem. Moreover, this sequence of feasible points is shown to converge in the sense of a weighted H ₂ norm; and it does so to the solution of the H ₂/H problem whenever such a solution exists.     

Convexification and Concavification for a General Class of Global Optimization Problems

A kind of general convexification and concavification methods is proposed for solving some classes of global optimization problems with certain monotone properties. It is shown that these minimization problems can be transformed into equivalent concave minimization problem or reverse convex programming problem or canonical D.C. programming problem by using the proposed convexification and concavification schemes. The existing algorithms then can be used to find the global solutions of the transformed problems.     

Multiplicative Programming Problems: Analysis and Efficient Point Search Heuristic

Multiplicative programming problems are difficult global optimization problems known to be NP-hard. At the same time, these problems have some important applications in engineering, finance, economics, and other fields. This article has two purposes. The first is to present an analysis that shows several relationships between concave multiplicative programs and concave minimization problems, and between concave multiplicative programs and certain multiple-objective mathematical programs. The second purpose is to propose and report computational results for a heuristic efficient-point search algorithm that we have designed for use on linear multiplicative programming problems. To our knowledge, this is the first heuristic algorithm of its type. The theoretical and algorithmic results given in the article offer some potentially important new avenues for analyzing and solving multiplicative programming problems of various types.     

<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Observer-Based Control for a Class of Uncertain Neutral Time-Delay Systems via LMI Optimization Approach

In this paper, the H_∞ observer-based control for a class of uncertain neutral time-delay systems is considered. The linear matrix inequality (LMI) optimization approach is used to design the H_∞ robust control with disturbance attenuation. Two classes of observer-based controls are proposed and their H_∞-norm bounds are given. The control and observer gains are given from the LMI feasible solutions. A numerical example is given to illustrate the results. The research reported here was supported by the National Science Council of Taiwan under Grant NSC 93-2213-E-214-020.     

Global minimization of a generalized convex multiplicative function

This paper discusses an algorithm for generalized convex multiplicative programming problems, a special class of nonconvex minimization problems in which the objective function is expressed as a sum ofp products of two convex functions. It is shown that this problem can be reduced to a concave minimization problem with only 2p variables. An outer approximation algorithm is proposed for solving the resulting problem.     

A general class of branch-and-bound methods in global optimization with some new approaches for concave minimization

Based on a review of existing algorithms, a general branch-and-bound concept in global optimization is presented. A sufficient and necessary convergence condition is established, and a broad class of realizations is derived that include existing and several new approaches for concave minimization problems.     

Comparative studies on dynamic programming and integer programming approaches for concave cost production/inventory control problems

This paper is concerned with classical concave cost multi-echelon production/inventory control problems studied by W. Zangwill and others. It is well known that the problem with m production steps and n time periods can be solved by a dynamic programming algorithm in O(n ⁴ m) steps, which is considered as the fastest algorithm for solving this class of problems. In this paper, we will show that an alternative 0–1 integer programming approach can solve the same problem much faster particularly when n is large and the number of 0–1 integer variables is relatively few. This class of problems include, among others problem with set-up cost function and piecewise linear cost function with fewer linear pieces. The new approach can solve problems with mixed concave/convex cost functions, which cannot be solved by dynamic programming algorithms.