Superoptimal Solution Research Articles

Complexity, exactness, and rationality in polynomial optimization

We focus on rational solutions or nearly-feasible rational solutions that serve as certificates of feasibility for polynomial optimization problems. We show that, under some separability conditions, certain cubic polynomially constrained sets admit rational solutions. However, we show in other cases that it is NP Hard to detect if rational solutions exist or if they exist of any reasonable size. We extend this idea to various settings including near feasible, but super optimal solutions and detecting rational rays on which a cubic function is unbounded. Lastly, we show that in fixed dimension, the feasibility problem over a set defined by polynomial inequalities is in NP by providing a simple certificate to verify feasibility. We conclude with several related examples of irrationality and encoding size issues in QCQPs and SOCPs.

Utopia in the solution of the Bucket Order Problem

This paper deals with group decision making and, in particular, with rank aggregation, which is the problem of aggregating individual preferences (rankings) in order to obtain a consensus ranking. Although this consensus ranking is usually a permutation of all the ranked items, in this paper we tackle the situation in which some items can be tied, that is, the consensus shows that there is no preference among them. This problem has arisen recently and is known as the Optimal Bucket Order Problem (OBOP).In this paper we propose two improvements to the standard greedy algorithm usually considered to approach the bucket order problem: the Bucket Pivot Algorithm (BPA). The first improvement is based on the introduction of the Utopian Matrix, a matrix associated to a pair order matrix that represents the precedences in a collection of rankings. This idealization constitutes a superoptimal solution to the OBOP, which can be used as an extreme (sometimes feasible) best value. The second improvement is based on the use of several items as pivots to generate the bucket order, in contrast to BPA that only uses a single pivot. The set of items playing the role of decision-maker is dynamically created. We analyze separately the contribution of each improvement and also their joint effect. The statistical analysis of the experiments carried out shows that the combined use of both techniques is the best choice, showing a significant improvement in accuracy (17%) with respect to the original BPA and providing an important reduction in the variance of the output. Moreover, we provide decision rules to help the decision maker to select the right algorithm according to the problem instance.

Solving integer programs over monotone inequalities in three variables: A framework for half integrality and good approximations

We define a class of monotone integer programs with constraints that involve up to three variables each. A generic constraint in such integer program is of the form ax− by⩽ z+ c, where a and b are nonnegative and the variable z appears only in that constraint. We devise an algorithm solving such problems in time polynomial in the length of the input and the range of variables U. The solution is obtained from a minimum cut on a graph with O( nU) nodes and O( mU) arcs where n is the number of variables of the types x and y and m is the number of constraints. Our algorithm is also valid for nonlinear objective functions. Nonmonotone integer programs are optimization problems with constraints of the type ax+ by⩽ z+ c without restriction on the signs of a and b. Such problems are in general NP-hard. We devise here an algorithm, relying on a transformation to the monotone case, that delivers half integral superoptimal solutions in polynomial time. Such solutions provide bounds on the optimum value that can only be superior to bounds provided by linear programming relaxation. When the half integral solution can be rounded to an integer feasible solution, this is a 2-approximate solution. In that the technique is a unified 2-approximation technique for a large class of problems. The results apply also for general integer programming problems with worse approximation factors that depend on a quantifier measuring how far the problem is from the class of problems we describe. The algorithm described here has a wide array of problem applications. An additional important consequence of our results is that nonmonotone problems in the framework are MAX SNP-hard and at least as hard to approximate as vertex cover. Problems that are amenable to the analysis provided here are easily recognized. The analysis itself is entirely technical and involves manipulating the constraints and transforming them to a totally unimodular system while losing no more than a factor of 2 in the integrality.

Approximation by Analytic Matrix Functions: The Four Block Problem

The four block problem is a generalization of Nehari's problem for matrix functions. It plays an important role inH∞-optimal control theory. It is well known that Nehari's problem for a continuous scalar function has a unique solution. However, in the matrix case the situation is quite different. V. V. Peller and N. J. Young (1994,J. Funct. Anal.120, 300–343) studiedsuperoptimal solutionsof Nehari's problem. They minimize not only theL∞-norm of the corresponding matrix function but also the essential suprema of all further singular values. It was shown that forH∞+Cmatrix functions Nehari's problem has a unique superoptimal solution. In this paper we study superoptimal solutions of the four block problem and we find a natural condition under which such a superoptimal solution is unique. Our result is new even in the case of Nehari's problem. We study some related problems such as thematic factorizations, invariance of indices, and inequalities between the singular values of the four block operator and the superoptimal singular values.

On Superoptimal Approximation by Analytic and Meromorphic Matrix-Valued Functions

It is a well-known fact that for any continuous scalar-valued function φ on the unit circle there is a unique best approximation in the Hardy class H ∞ of bounded analytic functions. However, in the matrix-valued case a best approximation by bounded analytic functions is almost never unique. To make it unique, one has to impose additional assumptions. In 1986 N. J. Young introduced the so-called superoptimal solution of the Nehari problem. The word superoptimal means that we are seeking F ∈ H ∞ (matrix-valued) to minimize not only sup {|| Φ (ζ) − F(ζ) || : |ζ| = 1} = sup{ s 0(Φ(ζ) − F (ζ)) : |ζ| = 1} but also the suprema of further singular values s j (Φ(ζ) − F(ζ)), j ≥ 1. It was proved by V. V. Peller and N. J. Young that for matrix-valued function Φ in H ∞ + C such superoptiminal solution is unique. In this paper an alternative geometric approach to the problem is presented. It allows us to obtain another proof of the uniqueness and some new results: uniqueness of the superoptimal approximation by meromorphic functions (a generalization of the Adamyan-Arov-Krein result on approximation of scalar-valued functions), inequalities between s-numbers of a Hankel operator, and superoptimal singular values. Our approach works well for operator-valued functions as for matrix-valued ones.

A Super-Optimization Method for Four-Block Problems

This paper deals with a discrete time frequency domain approach to super-optimization of four-block problems via a dimension reduction and diagonalization technique based on the so-called equalizer principle, an optimality criterion associated with the polynomial approach to $\mathcal{H}^\infty $ -optimization by Kwakernaak. The reduction technique provides a hierarchical decomposition of the super-optimization problem into successive ordinary $\mathcal{H}^\infty $-optimization problems, one for each singular value to be minimized. The procedure allows computation of super-optimal solutions for a large class of four-block problems.

A frame approach to the H/sub infinity / superoptimal solution

A frame approach to the H/sub infinity / superoptimal solution which offers computational improvements over existing algorithms is given. The approach is based on interpreting s numbers as the largest gains between appropriately defined spaces. Some useful bounds on Hankel singular values and s numbers are derived.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Improved formulae for the 2-block H∞ super-optimal solution

Improved formulae are presented for the algorithm given in Gu et al. (Automatica 25, 215–221 (1989) for finding 2-block H ∞ super-optimal solutions.

An algorithm for super-optimal H∞ design: The two-block case

In this paper, we consider the strengthened H ∞ optimization problem in the two-block case. An algorithm for the super-optimal solution is proposed. It is based on results by the authors ( IEEE Trans. Aut. Control, AC-33, 833 (1988)) for the less practical one-block case. It uses state-space models and is numerically implementable. A feature of the procedure is an optimality criterion which is a natural generalization of that used in the iterative scheme of Doyle (Lecture Notes in Advances in Multivariable Control, ONR/Honeywell Workshop, Minneapolis, U.S.A. (1984)) for the unstrengthened problem.

A computational algorithm for the super-optimal solution of the model matching problem

In this paper, we develop a computational algorithm to implement the strengthened minimization technique [1] in the model matching problem which arises in robust controller design and weighted sensitivity minimization. A general algorithm is proposed for constructing unitary matrices and the extremal functions for the H ∞-optimization problem are diagonalized using the constructed unitary matrices. We impose stronger minimization conditions to force the solution of the optimization problem to be unique and explain their physical significance.

A Polynomial Approximation Scheme for Scheduling on Uniform Processors: Using the Dual Approximation Approach

We present a polynomial approximation scheme for the minimum makespan problem on uniform parallel processors. More specifically, the problem is to find a schedule for a set of independent jobs on a collection of machines of different speeds so that the last job to finish is completed as quickly as possible. We give a family of polynomial-time algorithms $\{ {A_\varepsilon } \}$ such that $A_\varepsilon $ delivers a solution that is within a relative error $\varepsilon $ of the optimum. This is a dramatic improvement over previously known algorithms; the best performance guarantee previously proved for a polynomial-time algorithm ensured a relative error no more than 40 percent. The technique employed is the dual approximation approach, where infeasible but superoptimal solutions for a related (dual) problem are converted to the desired feasible but possibly suboptimal solution.