Approximate parametric optimization of infinite-dimensional systems

We consider the problem of finding g M_n such that where M_n is the n-dimensional subspace of the complexHilbert space L²(0, ) spanned by an n-tuple of normalized eigenvectoesof the operator , corresponding to eigenvalues. The solution is g = P_nf and P_n denotesthe orthoprojector onto M_n. From Grabowski (1991) we know thatP_n can be expressed in terms of the Malmquist functions. Wegive an alternative approach, more convenient for applicationof the standard mathematical software. The problem of convergenceas n is discussed from both theoretical and numerical viewpoint.The reslts are illustrated by the problems of finding the optimaladjustment of the proportional controller stabilizing a distributedplant. Email: pgrab{at}ia.agh.edu.pl     

A survey of vector optimization in infinite-dimensional spaces,part 2

The present survey deals with the state of vector optimization as a mathematical discipline. In this context, the optima are generally defined as maximal pointsy ₀ with respect to a partial order on the criteria space. The survey is restricted to a discussion of that literature which deals with pointsy ₀ which satisfy a maximality condition with respect toy ₀-comparable criteria values; papers which are based on a maximality condition satisfied for all admissible criteria values are included only in a supplementary bibliography. For the former, all aspects of the optimization process are surveyed, ranging from questions of existence to the treatment of duality. Particular attention is paid to questions of proper maximality. The discussion is based on a broad range of definitions and selected theorems from the literature.The authors wish to express their appreciation to O. Saleh and Y. H. Liu for their thoughtful ear, their cogent suggestions, and their untiring legwork in procuring and discussing many of the papers cited in this review. Above all, however, their thanks go to Professors L. Hurwicz and J. M. Borwein for the excellent papers they produced in this area. Their work was a pleasure to read, and it provided the supporting framework without which our task would have been a considerably more difficult one.     

The lagrange-newton method for infinite-dimensional optimization problems

This paper investigates local convergence properties of the Lagrange-Newton method for optimization problems in reflexive Banach spaces. Sufficient conditions for quadratic convergence of optimal solutions and Lagrange multipliers are given. The results are applied to optimal control problems.     

Compact optimization can outperform separation: A case study in structural proteomics

In Combinatorial Optimization, one is frequently faced with linear programming (LP) problems with exponentially many constraints, which can be solved either using separation or what we call compact optimization. The former technique relies on a separation algorithm, which, given a fractional solution, tries to produce a violated valid inequality. Compact optimization relies on describing the feasible region of the LP by a polynomial number of constraints, in a higher dimensional space. A commonly held belief is that compact optimization does not perform as well as separation in practice. In this paper, we report on an application in which compact optimization does in fact largely outperform separation. The problem arises in structural proteomics, and concerns the comparison of 3-dimensional protein folds. Our computational results show that compact optimization achieves an improvement of up to two orders of magnitude over separation. We discuss some reasons why compact optimization works in this case but not, e.g., for the LP relaxation of the TSP.Received: April 2003, Revised: January 2004, MSC classification: 65K05, 90C05, 90C90Thanks to Arie Tamir for pointing out to us many relevant references, and to Brian Walenz and Giulio Zanetti for helpful discussions. We thank two anonymous referees for their suggestions which improved this paper considerably. The second author wishes to thank Sandia National Labs, Albuquerque, NM, and the Mathematical Biosciences Institute, Columbus, OH, for their hospitality. Sandia is a multiprogram laboratory, operated by Sandia Corporation, a Lockhead Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000.     

On cutoff optimization methods in infinite-dimensional spaces and applications

A general formulation of cutoff optimization methods in infinite-dimensional spaces is given. Various convergence aspects of these methods are proved under general assumptions. The concept of strong and weak convergence is studied. Estimates of convergence rates under certain assumptions are obtained. Applications to optimal control theory and approximation theory are illustrated.The author gratefully acknowledges many helpful suggestions made by the referees.     

Infinite Penalization for Optimal Control Problems: An infinite-dimensional optimization method for constrained optimization problems

We present results on a method for infinite dimensional constrained optimization problems. In particular, we are interested in state constrained optimal control problems and discuss an algorithm based on penalization and smoothing. The algorithm contains update rules for the penalty and the smoothing parameter that depend on the constraint violation. Theoretical as well as numerical results are given. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Directional Lipschitzian optimal solution of infinite-dimensional optimization problems

This paper presents a study of the Lipschitz dependence of the optimal solution of elementary convex programs in a Hilbert space when the equality constraints are subjected to small perturbations in some fixed direction and with the sub- and super-quadratic growth conditions. This study follows the recent results of Janin and Gauvin [1] related to the finite-dimentional case. As an illustrative example, we study the directional derivative with respect to the boundary conditions of the infimum (value function) of the Mossolov problem in space dimension one.     

Deformation method for the investigation of infinite-dimensional optimization problems

Translated from Matematicheskie Zametki, Vol. 53, No. 2, pp. 175–176, February, 1993.     

An optimization method to estimate models with store-level data: A case study

The quality of the estimation of a latent segment model when only store-level aggregate data is available seems to be dependent on the computational methods selected and in particular on the optimization methodology used to obtain it. Following the stream of work that emphasizes the estimation of a segmentation structure with aggregate data, this work proposes an optimization method, among the deterministic optimization methods, that can provide estimates for segment characteristics as well as size, brand/product preferences and sensitivity to price and price promotion variation estimates that can be accommodated in dynamic models. It is shown that, among the gradient based optimization methods that were tested, the Sequential Quadratic Programming method (SQP) is the only that, for all scenarios tested for this type of problem, guarantees of reliability, precision and efficiency being robust, i.e., always able to deliver a solution. Therefore, the latent segment models can be estimated using the SQP method when only aggregate market data is available.     

An optimization approach for communal home meal delivery service: A case study

This paper is the first to discuss the communal home meal delivery problem. The problem can be modelled as a multiple travelling salesman problem with time windows, that is closely related to the well-studied vehicle routing problem with time windows. Experimental results are reported for a real-life case study from Central Finland over several alternative scenarios using the SPIDER commercial solver. The comparison with current practice reveals that a significant savings potential can be obtained using off-the-shelf optimization tools. As such, the potential for supporting real-life communal routing problems can be considered to be important for VRP practitioners.     

Matrix infinite-dimensional analog of the Wiener theorem on local inversion of Fourier transforms

A matrix infinite-dimensional analog of the Wiener theorem on local inversion of Fourier transforms is proved.Deceased.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 141–145, January, 1995.This research was supported by Ukrainian State Committee on Science and Technology.     

Hyper-spherical inversion transformations in multi-objective evolutionary optimization

Multi-objective evolutionary algorithms (MOEAs) are widely considered to have two goals: convergence towards the true Pareto front and maintaining a diverse set of solutions. The primary concern here is with the first goal of convergence, in particular when one or more variables must converge to a constant value. Using a number of well known test problems, the difficulties that are currently impeding convergence are discussed and then a new method is proposed that transforms the decision space using the geometric properties of hyper-spherical inversions to converge towards/onto the true Pareto front. Future extensions of this work and its application to multi-objective optimisation is discussed.     

Patient mix optimization in tactical cardiothoracic surgery planning: a case study

Cardiothoracic surgery planning involves different resourcessuch as operating theatre (OT) time, medium care beds, intensivecare beds and nursing staff. Within cardiothoracic surgery differentcategories of patients can be distinguished with respect totheir requirements of resources. The mix of patients is, therefore,an important aspect of decision making for the hospital to managethe use of these resources. A master OT schedule is used atthe tactical level of planning for deriving the weekly OT plan.It defines for each day of a week the number of OT hours availableand the number of patients operated from each patient category.We develop a model for this tactical level planning problem,the core of which is a mixed integer linear program. The modelis used to evaluate scenarios for surgery planning at tacticalas well as strategic levels, demonstrating the potential ofinteger programming for providing recommendations for change.     

Local optimization of black-box functions with high or infinite-dimensional inputs: application to nuclear safety

Black-box optimization problems when the input space is a high-dimensional space or a function space appear in more and more applications. In this context, the methods available for finite-dimensional data do not apply. The aim is then to propose a general method for optimization involving dimension reduction techniques. Different dimension reduction basis are considered (including data-driven basis). The methodology is illustrated on simulated functional data. The choice of the different parameters, in particular the dimension of the approximation space, is discussed. The method is finally applied to a problem of nuclear safety.     

A comparative analysis of linear fitting for non-linear functions on optimization: A case study: Air pollution problems

A very frequent problem in advanced mathematical programming models is the linear approximation of convex and non-convex non-linear functions in either the constraints or the objective function of an otherwise linear programming problem. In this paper, based on a model that has been developed for the evaluation and selection of pollutant emission control policies and standards, we shall study several ways of representing non-linear functions of a single argument in mixed integer, separable and related programming terms. Thus we shall study the approximations based on piecewise constant, piecewise adjacent, piecewise non-adjacent additional and piecewise non-adjacent segmented functions. In each type of modelization we show the problem size and optimization results of using the following techniques: separable programming, mixed integer programming with Special Ordered Sets of type 1, linear programming with Special Ordered Sets of type 2 and mixed integer programming using strategies based on the quasi-integrality of the binary variables.     

A remark on A-weakly infinite-dimensional spaces

It is shown that the product of two A-weakly infinite-dimensional spaces may fail to have this property and that under CH there is no universal space in the class of all metrizable separable A-weakly infinite-dimensional spaces.     

A general approach to infinite-dimensional holomorphy

In this paper we present a general theory for holomorphic functions which is based on continuous convergence instead of topologies. The theory can be applied to locally convex spaces and bornological spaces.