In this report we propose a new recursive matrix formulation of limited memory variable metric methods. This approach can be used for an arbitrary update from the Broyden class (and some other updates) and also for the approximation of both the Hessian matrix and its inverse. The new recursive formulation requires approximately $4mn$ multiplications and additions per iteration, so it is comparable with other efficient limited memory variable metric methods. Numerical experiments concerning Algorithm...

A reformulation of a mathematical program is a formulation which shares some properties with, but is in some sense better than, the original program. Reformulations are important with respect to the choice and efficiency of the solution algorithms; furthermore, it is desirable that reformulations can be carried out automatically. Reformulation techniques are widespread in mathematical programming but interestingly they have never been studied under a unified framework. This paper attempts to move...

The purpose of this paper is to apply second order $\eta$-approximation method introduced to optimization theory by Antczak [2] to obtain a new second order $\eta$-saddle point criteria for vector optimization problems involving second order invex functions. Therefore, a second order $\eta$-saddle point and the second order $\eta$-Lagrange function are defined for the second order $\eta$-approximated vector optimization problem constructed in this approach. Then, the equivalence between an (weak) efficient solution of the...

In this paper, by using the second order $\eta$-approximation method introduced by Antczak [3], new saddle point results are obtained for a nonlinear mathematical programming problem involving second order invex functions with respect to the same function $\eta$. Moreover, a second order $\eta$-saddle point and a second order $\eta$-Lagrange function are defined for the so-called second order $\eta$-approximated optimization problem constructed in this method. Then, the equivalence between an optimal solution in the original...

We study a continuous version of the capacity and flow assignment problem (CFA) where the design cost is combined with an average delay measure to yield a non convex objective function coupled with multicommodity flow constraints. A separable convexification of each arc cost function is proposed to obtain approximate feasible solutions within easily computable gaps from optimality. On the other hand, DC (difference of convex functions) programming can be used to compute accurate upper bounds and...

We study a continuous version of the capacity and flow assignment problem (CFA) where the design cost is combined with an average delay measure to yield a non convex objective function coupled with multicommodity flow constraints. A separable convexification of each arc cost function is proposed to obtain approximate feasible solutions within easily computable gaps from optimality. On the other hand, DC (difference of convex functions) programming can be used to compute accurate upper bounds and...

Let $aøplusb=max(a,b)$ and $aøtimesb=a+b$ for $a,b\in ℝ$. Max-algebra is an analogue of linear algebra developed on the pair of operations $(øplus,øtimes)$ extended to matrices and vectors. The system of equations $Aøtimesx=b$ and inequalities $Cøtimesxłeqd$ have each been studied in the literature. We consider a problem consisting of these two systems and present necessary and sufficient conditions for its solvability. We also develop a polynomial algorithm for solving max-linear program whose constraints are max-linear equations and inequalities.

In this paper, based on a generalized Karush-Kuhn-Tucker (KKT) method a modified recurrent neural network model for a class of non-convex quadratic programming problems involving a so-called $Z$-matrix is proposed. The basic idea is to express the optimality condition as a mixed nonlinear complementarity problem. Then one may specify conditions for guaranteeing the global solutions of the original problem by using results from the S-lemma. This process is proved by building up a dynamic system from...

In the paper, some sufficient optimality conditions for strict minima of order $m$ in constrained nonlinear mathematical programming problems involving (locally Lipschitz) $(F,\rho )$-convex functions of order $m$ are presented. Furthermore, the concept of strict local minimizer of order $m$ is also used to state various duality results in the sense of Mond-Weir and in the sense of Wolfe for such nondifferentiable optimization problems.

We consider the multiple ellipses detection problem on the basis of a data points set coming from a number of ellipses in the plane not known in advance, whereby an ellipse $E$ is viewed as a Mahalanobis circle with center $S$, radius $r$, and some positive definite matrix $\Sigma$. A very efficient method for solving this problem is proposed. The method uses a modification of the $k$-means algorithm for Mahalanobis-circle centers. The initial approximation consists of the set of circles whose centers are determined...

Parallel multi-deme genetic algorithms are especially advantageous because they allow reducing the time of computations and can perform a much broader search than single-population ones. However, their formal analysis does not seem to have been studied exhaustively enough. In this paper we propose a mathematical framework describing a wide class of island-like strategies as a stationary Markov chain. Our approach uses extensively the modeling principles introduced by Vose, Rudolph and their collaborators....

The paper focuses on the acceleration of the computer optimization of heat radiation intensity on the mould surface. The mould is warmed up by infrared heaters positioned above the mould surface, and in this way artificial leathers in the automotive industry are produced (e.g. for car dashboards). The presented heating model allows us to specify the position of infrared heaters over the mould to obtain approximately even heat radiation intensity on the whole mould surface. In this way we can obtain...

In this paper we give necessary and sufficient optimality conditions for a vector optimization problem over cones involving support functions in objective as well as constraints, using cone-convex and other related functions. We also associate a unified dual to the primal problem and establish weak, strong and converse duality results. A number of previously studied problems appear as special cases.