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Information about several papers discusses in 11th International Conference on Stochastic Programming which was held in Vienna, Austria on August 27, 2007 is presented. The conference was aimed at closing the gap between the field of stochastic programming and the area of robust optimization and optimization under uncertainty. The conference featured Arkadi Nemirovski, Darinka Dentcheva and Andrzej Ruszczynski.

A preface to the journal "Optimization" is presented.

A preface for the February 2006 issue of the periodical "Optimization" is presented.

By means of a conjugation scheme based on generalized convex conjugation theory instead of Fenchel conjugation, we build an alternative dual problem, using the perturbational approach, for a general optimization one defined on a separated locally convex topological space. Conditions guaranteeing strong duality for primal problems which are perturbed by continuous linear functionals and their respective dual problems, which is named stable strong duality, are established. In these conditions, the fact that the perturbation function is evenly convex will play a fundamental role. Stable strong duality will also be studied in particular for Fenchel and Lagrange primal–dual problems, obtaining a characterization for Fenchel case. [ABSTRACT FROM PUBLISHER]

A generalized form of vector optimization problems in complex space is considered, where both the real and the imaginary parts of the objective functions are taken into account. The efficient solutions are defined and characterized in terms of optimal solutions of related appropriate scalar optimization problems. These scalar problems are formulated by means of vectors in the dual of the domination cone. Under analyticity hypotheses about the functions, complex extensions to necessary and sufficient conditions for efficiency of Kuhn–Tucker type are established. Most of the corresponding results of previous studies (in both finite-dimensional complex and real spaces) can be recovered as particular cases. [ABSTRACT FROM PUBLISHER]

We present two generalized conjugation schemes for lower semi-continuous functions defined on a real Banach space whose norm is Fréchet differentiable off the origin, and sketch their applications to optimization duality theory. Both approaches are based upon a new characterization of lower semi-continuous functions as pointwise suprema of a special class of continuous functions. [ABSTRACT FROM AUTHOR]

An infeasible interior-point method (IIPM) for solving linear optimization problems based on a kernel function with trigonometric barrier term is analysed. In each iteration, the algorithm involves a feasibility step and several centring steps. The centring step is based on classical Newton’s direction, while we used a kernel function with trigonometric barrier term in the algorithm to induce the feasibility step. The complexity result coincides with the best-known iteration bound for IIPMs. To our knowledge, this is the first full-Newton step IIPM based on a kernel function with trigonometric barrier term. [ABSTRACT FROM AUTHOR]

Second-order local optimality conditions involving copositivity of the Hessian of the Lagrangian on the reduced linearization cone have the advantage that there is only a small gap between sufficient (the Hessian is strictly copositive) and necessary (the Hessian is copositive) conditions. In this respect, this is a proper generalization of convexity of the Lagrangian. We also specify a copositivity-based variant which is sufficient for global optimality. For (non-convex) quadratic optimization problems over polyhedra (QPs), the distinction between sufficiency and necessity vanishes, both for local and global optimality. However, in the strictly copositive case we can provide a distance lower (error) bound of the incrementaround a local minimizer. This is a refinement of an earlier result which focussed on mere (non-strict) copositivity. In addition, an apparently new variant of constraint qualification (CQ) is presented which is implied by Abadie’s CQ and which is suitable for second-order analysis. This newreflected Abadie CQis neither implied, nor implies, Guignard’s CQ. However, it implies the necessary second-order local optimality condition based on copositivity. Application to the trust-region problem and several (counter) examples illustrates the advantage of this approach. [ABSTRACT FROM AUTHOR]

We investigate regularity conditions in optimal control problems with mixed constraints of a general geometric type, in which a closed non-convex constraint set appears. A closely related question to this issue concerns the derivation of necessary optimality conditions under some regularity conditions on the constraints. By imposing strong and weak regularity condition on the constraints, we provide necessary optimality conditions in the form of Pontryagin maximum principle for the control problem with mixed constraints. The optimality conditions obtained here turn out to be more general than earlier results even in the case when the constraint set is convex. The proofs of our main results are based on a series of technical lemmas which are gathered in the Appendix. [ABSTRACT FROM AUTHOR]

We revisit the problem of integrability in the consumer theory, focusing on the main difficulties. First, we look for a neat and simple local existence result, and then for a global solution. Second, observing that a utility function (or indirect utility function) cannot be determined uniquely, we propose a means to get a kind of uniqueness result. Our approach is coordinate-free and can be used both in the classical case of a finite-dimensional commodity space and in the case an infinite-dimensional model is adopted. [ABSTRACT FROM AUTHOR]

In this article, we present an improved three-term conjugate gradient algorithm for large-scale unconstrained optimization. The search directions in the developed algorithm are proved to satisfy an approximate secant equation as well as the Dai-Liao’s conjugacy condition. With the standard Wolfe line search and the restart strategy, global convergence of the algorithm is established under mild conditions. By implementing the algorithm to solve 75 benchmark test problems with dimensions from 1000 to 10,000, the obtained numerical results indicate that the algorithm outperforms the state-of-the-art algorithms available in the literature. It costs less CPU time and smaller number of iterations in solving the large-scale unconstrained optimization. [ABSTRACT FROM AUTHOR]

We deliver formulae for the biconjugate functions of some infimal functions that hold provided the fulfilment of weak regularity conditions of both closedness and interiority types. As special cases, we obtain biconjugates of infimal convolutions of finitely many functions, of optimal value functions of both constrained and unconstrained optimization problems as well as of marginal functions associated with multifunctions (that can be, for instance, convex processes) and some given functions. Moreover, we rediscover or extend different results on biconjugate functions from the literature. [ABSTRACT FROM AUTHOR]

We show how to use intensively local cone approximations to obtain results in some fields of optimization theory as optimality conditions, constraint qualifications, mean value theorems and error bound. [ABSTRACT FROM AUTHOR]

The worst-case evaluation complexity of finding an approximate first-order critical point using gradient-related non-monotone methods for smooth non-convex and unconstrained problems is investigated. The analysis covers a practical linesearch implementation of these popular methods, allowing for an unknown number of evaluations of the objective function (and its gradient) per iteration. It is shown that this class of methods shares the known complexity properties of a simple steepest-descent scheme and that an approximate first-order critical point can be computed in at most) function and gradient evaluations, whereis the user-defined accuracy threshold on the gradient norm. [ABSTRACT FROM PUBLISHER]

We provide two weakly convergent algorithms for finding a zero of the sum of a maximally monotone operator, a cocoercive operator, and the normal cone to a closed vector subspace of a real Hilbert space. The methods exploit the intrinsic structure of the problem by activating explicitly the cocoercive operator in the first step, and taking advantage of a vector space decomposition in the second step. The second step of the first method is a Douglas–Rachford iteration involving the maximally monotone operator and the normal cone. In the second method, it is a proximal step involving the partial inverse of the maximally monotone operator with respect to the vector subspace. Connections between the proposed methods and other methods in the literature are provided. Applications to monotone inclusions with finitely many maximally monotone operators and optimization problems are examined. [ABSTRACT FROM PUBLISHER]

We examine a multidimensional optimization problem in the tropical mathematics setting. The problem involves the minimization of a non-linear function defined on a finite-dimensional semimodule over an idempotent semifield subject to linear inequality constraints. We start with an overview of known tropical optimization problems with linear and non-linear objective functions. A short introduction to tropical algebra is provided to offer a formal framework for solving the problem under study. As a preliminary result, a solution to a linear inequality with an arbitrary matrix is presented. We describe an example optimization problem drawn from project scheduling and then offer a general representation of the problem. To solve the problem, we introduce an additional variable and reduce the problem to the solving of a linear inequality, in which the variable plays the role of a parameter. A necessary and sufficient condition for the inequality to hold is used to evaluate the parameter, whereas the solution to the inequality is considered a solution to the problem. Based on this approach, a complete direct solution in a compact vector form is derived for the optimization problem under fairly general conditions. Numerical and graphical examples for two-dimensional problems are given to illustrate the obtained results. [ABSTRACT FROM PUBLISHER]