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.     

Asymptotic behavior of tails and quantiles of quadratic forms of Gaussian vectors

We derive results on the asymptotic behavior of tails and quantiles of quadratic forms of Gaussian vectors. They appear in particular in delta–gamma models in financial risk management approximating portfolio returns. Quantile estimation corresponds to the estimation of the Value-at-Risk, which is a serious problem in high dimension.     

SEQUENTIAL QUADRATIC PROGRAMMING METHODS FOR OPTIMAL CONTROL PROBLEMS WITH STATE CONSTRAINTS

A Kind of direct methods is presented for the solution of optimal control problems with state constraints.These methods are sequential quadratic programming methods.At every iteration a quadratic programming which is obtained by quadratic approximation to Lagrangian function and Linear approximations to constraints is solved to get a search direction for a merit function.The merit function is formulated by augmenting the Lagrangian funetion with a penalty term.A line search is carried out along the search direction to determine a step length such that the merit function is decreased.The methods presented in this paper include continuous sequential quadratic programming methods and discreate sequential quadrade programming methods.     

群体决策的偏好协调性检验

对于一类群体决策问题，本文引进群体的偏好协调性指标，并且给出了偏好协调性指标的统计检验．在此基础上，还提出一个求该类群体决策问题的方法，以及讨论了群体的偏好快调性指标和群体决策结果间的关系．     

On a Quadratic Optimization Problem with Equality Constraints

The constrained optimization problem with a quadratic cost functional and two quadratic equality constraints has been studied by Bar-on and Grasse, with positive-definite matrix in the objective. In this note, we shall relax the matrix in the objective to be positive semidefinite. A necessary and sufficient condition to characterize a local optimal solution to be global is established. Also, a perturbation scheme is proposed to solve this generalized problem.     

An improved linearization strategy for zero-one quadratic programming problems

We present a new linearized model for the zero-one quadratic programming problem, whose size is linear in terms of the number of variables in the original nonlinear problem. Our derivation yields three alternative reformulations, each varying in model size and tightness. We show that our models are at least as tight as the one recently proposed in [7], and examine the theoretical relationship of our models to a standard linearization of the zero-one quadratic programming problem. Finally, we demonstrate the efficacy of solving each of these models on a set of randomly generated test instances.     

COSEARCH: A Parallel Cooperative Metaheuristic

In order to design a well-balanced metaheuristic for robustness, we propose the COSEARCH approach which manages the cooperation of complementary heuristic methods via an adaptive memory which contains a history of the search already done. In this paper, we present the idiosyncrasies of the COSEARCH approach and its application for solving large scale instances of the quadratic assignment problem (QAP). We propose an original design of the adaptive memory in order to focus on high quality regions of the search and avoid attractive but deceptive areas. For the QAP, we have hybridized three heuristic agents of complementary behaviours: a Tabu Search is used as the main search algorithm, a Genetic Algorithm is in charge of the diversification and a Kick Operator is applied to intensify the search. The evaluations have been executed on large scale network of workstations via a parallel environment which supports fault tolerance and adaptive dynamic scheduling of tasks.     

Some proximity and sensitivity results in quadratic integer programming

We show that for any optimal solution for a given separable quadratic integer programming problem there exist an optimal solution for its continuous relaxation such that wheren is the number of variables and(A) is the largest absolute subdeterminant of the integer constraint matrixA. Also for any feasible solutionz, which is not optimal for the separable quadratic integer programming problem, there exists a feasible solution having greater objective function value and with . We further prove, under some additional assumptions, that the distance between a pair of optimal solutions to an integer quadratic programming problem with right hand side vectorsb andb, respectively, depends linearly on b–b₁. Finally the validity of all the results for nonseparable mixed-integer quadratic programs is established. The proximity results obtained in this paper are extensions of some of the results described in Cook et al. (1986) for linear integer programming.This research was partially supported by Natural Sciences and Engineering Research Council of Canada Grant 5-83998.