Asymptotic Properties of Random Multidimensional Assignment Problems

The multidimensional assignment problem (MAP) is a NP-hard combinatorial optimization problem, occurring in many applications, such as data association. In this paper, we prove two conjectures made in Ref. 1 and based on data from computational experiments on MAPs. We show that the mean optimal objective function cost of random instances of the MAP goes to zero as the problem size increases, when assignment costs are independent exponentially or uniformly distributed random variables. We prove also that the mean optimal solution goes to negative infinity when assignment costs are independent normally distributed random variables.     

Branch and win: OR tree search algorithms for solving combinatorial optimisation problems

Currently, most combinatorial optimisation problems have to be solved, if the optimum solution is sought, using general techniques to explore the space of feasible solutions and, more specifically, through exploratory enumerative procedures in trees and search graphs. We propose Branch and Win, a general formulation for understanding and synthesising the different tree search procedures that have been presented in the literature of operations research as well as in that of artificial intelligence. Several general ideas are also presented, whose application allows designing new hybrid search algorithms, in order to implement the procedure.     

Algorithms for Finite and Semi-Infinite Min–Max–Min Problems Using Adaptive Smoothing Techniques

We develop two implementable algorithms, the first for the solution of finite and the second for the solution of semi-infinite min-max-min problems. A smoothing technique (together with discretization for the semi-infinite case) is used to construct a sequence of approximating finite min-max problems, which are solved with increasing precision. The smoothing and discretization approximations are initially coarse, but are made progressively finer as the number of iterations is increased. This reduces the potential ill-conditioning due to high smoothing precision parameter values and computational cost due to high levels of discretization. The behavior of the algorithms is illustrated with three semi-infinite numerical examples.     

On the Image Regularity Condition

In this paper, we continue the analysis of the image regularity condition (IRC) as introduced in a previous paper where we have proved that IRC implies the existence of generalized Lagrange-John multipliers with first component equal to 1. The term generalized is connected with the fact that the separation (in the image space) is not necessarily linear (when we have classic Lagrange-John multipliers), but it can be also nonlinear. Here, we prove that the IRC guarantees, also in the nondifferentiable case, the fact that 0 is a solution of the first-order homogeneized (linearized) problem obtained by means of the Dini-Hadamard derivatives.     

Finding Robust Solutions Using Local Search

This paper investigates how a local search metaheuristic for continuous optimisation can be adapted so that it finds broad peaks, corresponding to robust solutions. This is relevant in problems in which uncertain or noisy data is present. When using a genetic or evolutionary algorithm, it is standard practice to perturb solutions once before evaluating them, using noise from a given distribution. This approach however, is not valid when using population-less techniques like local search and other heuristics that use local search. For those algorithms to find robust solutions, each solution needs to be perturbed and evaluated several times, and these evaluations need to be combined into a measure of robustness. In this paper, we examine how many of these evaluations are needed to reliably find a robust solution. We also examine the effect of the parameters of the noise distribution. Using a simple tabu search procedure, the proposed approach is tested on several functions found in the literature. This revised version was published online in August 2006 with corrections to the Cover Date.     

An existence theorem for neutral variational problems of Bolza

This note proves an existence theorem for a generalized Bolza-type problem that has time delays in both the state and velocity variables. The assumptions are stated in terms of a modification of the classical Hamiltonian, and extend ideas of Rockafellar to the delay case.     

Proper Efficiency in Vector Optimization on Real Linear Spaces

In this paper, we use an algebraic type of closure, which is called vector closure, and through it we introduce some adaptations to the proper efficiency in the sense of Hurwicz, Benson, and Borwein in real linear spaces without any particular topology. Scalarization, multiplier rules, and saddle-point theorems are obtained in order to characterize the proper efficiency in vector optimization with and without constraints. The usual convexlikeness concepts used in such theorems are weakened through the vector closure.     

Simultaneous Pseudo-Timestepping for PDE-Model Based Optimization Problems

In this paper we present a new method for the solution of optimization problems with PDE constraints. It is based on simultaneous pseudo-time stepping for evolution equations. The new method can be viewed as a continuous reduced SQP method in the sense that it uses a preconditioner derived from that method. The reduced Hessian in the preconditioner is approximated by a pseudo-differential operator, whose symbol can be investigated analytically. We apply our method to a boundary control model problem. The new optimization method needs 3.2-times the overall computational effort of the solution of simulation problem alone.     

The Module Structure Theorem for Sobolev Spaces on Open Manifolds

We prove a general embedding theorem for Sobolev spaces on open manifolds of bounded geometry and infer from this the module structure theorem. Thereafter we apply this to weighted Sobolev spaces.     

Interpolation by a Game

We introduce a notion of a real game (a generalisation of the Karchmer-Wigderson game (cf. [3]) and of real communication complexity, and relate this complexity to the size of monotone real formulas and circuits. We give an exponential lower bound for tree-like monotone protocols (defined in [4, Definition 2.2]) of small real communication complexity solving the monotone communication complexity problem associated with the bipartite perfect matching problem. This work is motivated by a research in interpolation theorems for prepositional logic (by a problem posed in [5, Section 8], in particular). Our main objective is to extend the communication complexity approach of [4, 5] to a wider class of proof systems. In this direction we obtain an effective interpolation in a form of a protocol of small real communication complexity. Together with the above mentioned lower bound for tree-like protocols this yields as a corollary a lower bound on the number of steps for particular semantic derivations of Hall's theorem (these include tree-like cutting planes proofs for which an exponential lower bound was demonstrated in [2]).