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On constant-weight TSP-tours

Scott Jones, P. Mark Kayll, Bojan Mohar, Walter D. Wallis (2003)

Discussiones Mathematicae Graph Theory

Is it possible to label the edges of Kₙ with distinct integer weights so that every Hamilton cycle has the same total weight? We give a local condition characterizing the labellings that witness this question's perhaps surprising affirmative answer. More generally, we address the question that arises when "Hamilton cycle" is replaced by "k-factor" for nonnegative integers k. Such edge-labellings are in correspondence with certain vertex-labellings, and the link allows us to determine (up to a constant...

On constraint qualifications in directionally differentiable multiobjective optimization problems

Giorgio Giorgi, Bienvenido Jiménez, Vincente Novo (2004)

RAIRO - Operations Research - Recherche Opérationnelle

We consider a multiobjective optimization problem with a feasible set defined by inequality and equality constraints such that all functions are, at least, Dini differentiable (in some cases, Hadamard differentiable and sometimes, quasiconvex). Several constraint qualifications are given in such a way that generalize both the qualifications introduced by Maeda and the classical ones, when the functions are differentiable. The relationships between them are analyzed. Finally, we give several Kuhn-Tucker...

On constraint qualifications in directionally differentiable multiobjective optimization problems

Giorgio Giorgi, Bienvenido Jiménez, Vincente Novo (2010)

RAIRO - Operations Research

We consider a multiobjective optimization problem with a feasible set defined by inequality and equality constraints such that all functions are, at least, Dini differentiable (in some cases, Hadamard differentiable and sometimes, quasiconvex). Several constraint qualifications are given in such a way that generalize both the qualifications introduced by Maeda and the classical ones, when the functions are differentiable. The relationships between them are analyzed. Finally, we give several Kuhn-Tucker...

On continuous convergence and epi-convergence of random functions. Part I: Theory and relations

Silvia Vogel, Petr Lachout (2003)

Kybernetika

Continuous convergence and epi-convergence of sequences of random functions are crucial assumptions if mathematical programming problems are approximated on the basis of estimates or via sampling. The paper investigates “almost surely” and “in probability” versions of these convergence notions in more detail. Part I of the paper presents definitions and theoretical results and Part II is focused on sufficient conditions which apply to many models for statistical estimation and stochastic optimization....

On continuous convergence and epi-convergence of random functions. Part II: Sufficient conditions and applications

Silvia Vogel, Petr Lachout (2003)

Kybernetika

Part II of the paper aims at providing conditions which may serve as a bridge between existing stability assertions and asymptotic results in probability theory and statistics. Special emphasis is put on functions that are expectations with respect to random probability measures. Discontinuous integrands are also taken into account. The results are illustrated applying them to functions that represent probabilities.

On convergence of the empirical mean method for non-identically distributed random vectors

E. Gordienko, J. Ruiz de Chávez, E. Zaitseva (2014)

Applicationes Mathematicae

We consider the following version of the standard problem of empirical estimates in stochastic optimization. We assume that the underlying random vectors are independent and not necessarily identically distributed but that they satisfy a "slow variation" condition in the sense of the definition given in this paper. We show that these assumptions along with the usual restrictions (boundedness and equicontinuity) on a class of functions allow one to use the empirical mean method to obtain a consistent...

On cycling in the simplex method of the transportation problem

Włodzimierz Szwarc (2009)

Applicationes Mathematicae

This paper shows that cycling of the simplex method for the m × n transportation problem where k-1 zero basic variables are leaving and reentering the basis does not occur once it does not occur in the k × k assignment problem. A method to disprove cycling for a particular k is applied for k=2,3,4,5 and 6.

On designing connected rapid transit networks reducing the number of transfers

Laureano Fernando Escudero, Susana Muñoz (2011)

RAIRO - Operations Research - Recherche Opérationnelle

In this paper we introduce some improvements on an approach that we described elsewhere for solving a modification of the well-known extended rapid transit network design problem. Firstly, we propose an integer programming model for selecting the stations to be constructed and the links between them, in such a way that a connected rapid transit network is obtained. Secondly, we consider a linear 0-1 programming model for determining a route of minimum length in the rapid transit network between...

On designing connected rapid transit networks reducing the number of transfers

Laureano Fernando Escudero, Susana Muñoz (2012)

RAIRO - Operations Research

In this paper we introduce some improvements on an approach that we described elsewhere for solving a modification of the well-known extended rapid transit network design problem. Firstly, we propose an integer programming model for selecting the stations to be constructed and the links between them, in such a way that a connected rapid transit network is obtained. Secondly, we consider a linear 0-1 programming model for determining a route of minimum...

On dual vector optimization and shadow prices

Letizia Pellegrini (2004)

RAIRO - Operations Research - Recherche Opérationnelle

In this paper we present the image space analysis, based on a general separation scheme, with the aim of studying lagrangian duality and shadow prices in Vector Optimization. Two particular kinds of separation are considered; in the linear case, each of them is applied to the study of sensitivity analysis, and it is proved that the derivatives of the perturbation function can be expressed in terms of vector Lagrange multipliers or shadow prices.

On dual vector optimization and shadow prices

Letizia Pellegrini (2010)

RAIRO - Operations Research

In this paper we present the image space analysis, based on a general separation scheme, with the aim of studying Lagrangian duality and shadow prices in Vector Optimization. Two particular kinds of separation are considered; in the linear case, each of them is applied to the study of sensitivity analysis, and it is proved that the derivatives of the perturbation function can be expressed in terms of vector Lagrange multipliers or shadow prices.

On existence of solutions to degenerate nonlinear optimization problems

Agnieszka Prusińska, Alexey Tret'yakov (2007)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We investigate the existence of the solution to the following problem min φ(x) subject to G(x)=0, where φ: X → ℝ, G: X → Y and X,Y are Banach spaces. The question of existence is considered in a neighborhood of such point x₀ that the Hessian of the Lagrange function is degenerate. There was obtained an approximation for the distance of solution x* to the initial point x₀.

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