Stability in Linear Optimization Under Perturbations of the Left-Hand Side Coefficients

This paper studies stability properties of linear optimization problems with finitely many variables and an arbitrary number of constraints, when only left hand side coefficients can be perturbed. The coefficients of the constraints are assumed to be continuous functions with respect to an index which ranges on certain compact Hausdorff topological space, and these properties are preserved by the admissible perturbations. More in detail, the paper analyzes the continuity properties of the feasible set, the optimal set and the optimal value, as well as the preservation of desirable properties (boundedness, uniqueness) of the feasible and of the optimal sets, under sufficiently small perturbations.

Vector Combinatorial Problems in a Space of Combinations with Linear Fractional Functions of Criteria

The paper considers vector discrete optimization problem with linear fractional functions of criteria on a feasible set that has combinatorial properties of combinations. Structural properties of a feasible solution domain and of Pareto–optimal (efficient), weakly efficient, strictly efficient solution sets are examined. A relation between vector optimization problems on a combinatorial set of combinations and on a continuous feasible set is determined. One possible approach is proposed in order to solve a multicriteria combinatorial problem with linear- fractional functions of criteria on a set of combinations.

Strong-weak Stackelberg Problems in Finite Dimensional Spaces

We are concerned with two-level optimization problems called strongweak Stackelberg problems, generalizing the class of Stackelberg problems in the strong and weak sense. In order to handle the fact that the considered two-level optimization problems may fail to have a solution under mild assumptions, we consider a regularization involving ε-approximate optimal solutions in the lower level problems. We prove the existence of optimal solutions for such regularized problems and present some approximation results when the parameter ǫ goes to zero. Finally, as an example, we consider an optimization problem associated to a best bound given in [2] for a system of nondifferentiable convex inequalities.

Generalized Scalarizing Problems GENS and GENSLex of Multicriteria Optimization

* This paper is partially supported by the National Science Fund of Bulgarian Ministry of Education and Science under contract № I–1401\2004 "Interactive Algorithms and Software Systems Supporting Multicriteria Decision Making".

On Pareto Sets in Multi-Criteria Optimization

Здравко Д. Славов - В тази работа се разглеждат Паретовските решения в непрекъсната многокритериална оптимизация. Обсъжда се ролята на някои предположения, които влияят на характеристиките на Паретовските множества. Авторът се е опитал да премахне предположенията за вдлъбнатост на целевите функции и изпъкналост на допустимата област, които обикновено се използват в многокритериалната оптимизация. Резултатите са на базата на конструирането на ретракция от допустимата област върху Парето-оптималното множество.

Efficient algorithms to solve scheduling problems with a variety of optimization criteria

La programmation par contraintes est une technique puissante pour résoudre, entre autres, des problèmes d’ordonnancement de grande envergure. L’ordonnancement vise à allouer dans le temps des tâches à des ressources. Lors de son exécution, une tâche consomme une ressource à un taux constant. Généralement, on cherche à optimiser une fonction objectif telle la durée totale d’un ordonnancement. Résoudre un problème d’ordonnancement signifie trouver quand chaque tâche doit débuter et quelle ressource doit l’exécuter. La plupart des problèmes d’ordonnancement sont NP-Difficiles. Conséquemment, il n’existe aucun algorithme connu capable de les résoudre en temps polynomial. Cependant, il existe des spécialisations aux problèmes d’ordonnancement qui ne sont pas NP-Complet. Ces problèmes peuvent être résolus en temps polynomial en utilisant des algorithmes qui leur sont propres. Notre objectif est d’explorer ces algorithmes d’ordonnancement dans plusieurs contextes variés. Les techniques de filtrage ont beaucoup évolué dans les dernières années en ordonnancement basé sur les contraintes. La proéminence des algorithmes de filtrage repose sur leur habilité à réduire l’arbre de recherche en excluant les valeurs des domaines qui ne participent pas à des solutions au problème. Nous proposons des améliorations et présentons des algorithmes de filtrage plus efficaces pour résoudre des problèmes classiques d’ordonnancement. De plus, nous présentons des adaptations de techniques de filtrage pour le cas où les tâches peuvent être retardées. Nous considérons aussi différentes propriétés de problèmes industriels et résolvons plus efficacement des problèmes où le critère d’optimisation n’est pas nécessairement le moment où la dernière tâche se termine. Par exemple, nous présentons des algorithmes à temps polynomial pour le cas où la quantité de ressources fluctue dans le temps, ou quand le coût d’exécuter une tâche au temps t dépend de t.

Distributed optimization in transportation and logistics networks

Many important problems in communication networks, transportation networks, and logistics networks are solved by the minimization of cost functions. In general, these can be complex optimization problems involving many variables. However, physicists noted that in a network, a node variable (such as the amount of resources of the nodes) is connected to a set of link variables (such as the flow connecting the node), and similarly each link variable is connected to a number of (usually two) node variables. This enables one to break the problem into local components, often arriving at distributive algorithms to solve the problems. Compared with centralized algorithms, distributed algorithms have the advantages of lower computational complexity, and lower communication overhead. Since they have a faster response to local changes of the environment, they are especially useful for networks with evolving conditions. This review will cover message-passing algorithms in applications such as resource allocation, transportation networks, facility location, traffic routing, and stability of power grids.

Optimal strategies and minimax lower bounds for online convex games

A number of learning problems can be cast as an Online Convex Game: on each round, a learner makes a prediction x from a convex set, the environment plays a loss function f, and the learner’s long-term goal is to minimize regret. Algorithms have been proposed by Zinkevich, when f is assumed to be convex, and Hazan et al., when f is assumed to be strongly convex, that have provably low regret. We consider these two settings and analyze such games from a minimax perspective, proving minimax strategies and lower bounds in each case. These results prove that the existing algorithms are essentially optimal.

More than merely wicked : regulating sewage discharges and other unspeakable problems in environmental policy-making and planning

While the communicative turn in policy-making has encouraged the public deliberation of policy decisions it has arguably had a more limited impact on the ability of public processes to deal with wicked problems. Wicked policy problems are characterised by high levels of complexity, uncertainty and divergence of values. However, some wicked problems present the additional challenge of high levels of psychosocial sensitivity and verbal proscription. Because these unspeakable policy problems frequently involve a significant moral dimension, the regulation of intimate processes or bodies, and strong elements of abjection and symbolic pollution they are quite literally problems that we don’t like to think about or talk about. However, the potential environmental and social impacts of these problems require that they be addressed. In this paper I present the preliminary findings of a research project focussed on the idea of the unspeakable policy problem and how its unspeakable nature can impact upon public participation and policy and environmental outcomes.

Evolutionary optimization and game strategies for advanced multi-disciplinary design : applications to aeronautics and UAV design

Many complex aeronautical design problems can be formulated with efficient multi-objective evolutionary optimization methods and game strategies. This book describes the role of advanced innovative evolution tools in the solution, or the set of solutions of single or multi disciplinary optimization. These tools use the concept of multi-population, asynchronous parallelization and hierarchical topology which allows different models including precise, intermediate and approximate models with each node belonging to the different hierarchical layer handled by a different Evolutionary Algorithm. The efficiency of evolutionary algorithms for both single and multi-objective optimization problems are significantly improved by the coupling of EAs with games and in particular by a new dynamic methodology named “Hybridized Nash-Pareto games”. Multi objective Optimization techniques and robust design problems taking into account uncertainties are introduced and explained in detail. Several applications dealing with civil aircraft and UAV, UCAV systems are implemented numerically and discussed. Applications of increasing optimization complexity are presented as well as two hands-on test cases problems. These examples focus on aeronautical applications and will be useful to the practitioner in the laboratory or in industrial design environments. The evolutionary methods coupled with games presented in this volume can be applied to other areas including surface and marine transport, structures, biomedical engineering, renewable energy and environmental problems.

Reliability Optimization using SLUMT

This paper is concerned with the reliability optimization of a spatially redundant system, subject to various constraints, by using nonlinear programming. The constrained optimization problem is converted into a sequence of unconstrained optimization problems by using a penalty function. The new problem is then solved by the conjugate gradient method. The advantages of this method are highlighted.

Non-linear Transceiver Designs with Imperfect CSIT Using Convex Optimization

In this paper, we consider non-linear transceiver designs for multiuser multi-input multi-output (MIMO) down-link in the presence of imperfections in the channel state information at the transmitter (CSIT). The base station (BS) is equipped with multiple transmit antennas and each user terminal is equipped with multiple receive antennas. The BS employs Tomlinson-Harashima precoding (THP) for inter-user interference pre-cancellation at the transmitter. We investigate robust THP transceiver designs based on the minimization of BS transmit power with mean square error (MSE) constraints, and balancing of MSE among users with a constraint on the total BS transmit power. We show that these design problems can be solved by iterative algorithms, wherein each iteration involves a pair of convex optimization problems. The robustness of the proposed algorithms to imperfections in CSIT is illustrated through simulations.

The alpha method for solving differential algebraic inequality (DAI) systems

This paper describes an algorithm for ``direct numerical integration'' of the initial value Differential-Algebraic Inequalities (DAI) in a time stepping fashion using a sequential quadratic programming (SQP) method solver for detecting and satisfying active path constraints at each time step. The activation of a path constraint generally increases the condition number of the active discretized differential algebraic equation's (DAE) Jacobian and this difficulty is addressed by a regularization property of the alpha method. The algorithm is locally stable when index 1 and index 2 active path constraints and bounds are active. Subject to available regularization it is seen to be stable for active index 3 active path constraints in the numerical examples. For the high index active path constraints, the algorithm uses a user-selectable parameter to perturb the smaller singular values of the Jacobian with a view to reducing the condition number so that the simulation can proceed. The algorithm can be used as a relatively cheaper estimation tool for trajectory and control planning and in the context of model predictive control solutions. It can also be used to generate initial guess values of optimization variables used as input to inequality path constrained dynamic optimization problems. The method is illustrated with examples from space vehicle trajectory and robot path planning.

Accurate Parameter Estimation of Contaminant Transport Inverse Problem using Particle Swarm Optimization

Swarm Intelligence techniques such as particle swarm optimization (PSO) are shown to be incompetent for an accurate estimation of global solutions in several engineering applications. This problem is more severe in case of inverse optimization problems where fitness calculations are computationally expensive. In this work, a novel strategy is introduced to alleviate this problem. The proposed inverse model based on modified particle swarm optimization algorithm is applied for a contaminant transport inverse model. The inverse models based on standard-PSO and proposed-PSO are validated to estimate the accuracy of the models. The proposed model is shown to be out performing the standard one in terms of accuracy in parameter estimation. The preliminary results obtained using the proposed model is presented in this work.

Pareto-optimal solutions for multi-objective optimization of fed-batch bioreactors using nondominated sorting genetic algorithm

Many optimal control problems are characterized by their multiple performance measures that are often noncommensurable and competing with each other. The presence of multiple objectives in a problem usually give rise to a set of optimal solutions, largely known as Pareto-optimal solutions. Evolutionary algorithms have been recognized to be well suited for multi-objective optimization because of their capability to evolve a set of nondominated solutions distributed along the Pareto front. This has led to the development of many evolutionary multi-objective optimization algorithms among which Nondominated Sorting Genetic Algorithm (NSGA and its enhanced version NSGA-II) has been found effective in solving a wide variety of problems. Recently, we reported a genetic algorithm based technique for solving dynamic single-objective optimization problems, with single as well as multiple control variables, that appear in fed-batch bioreactor applications. The purpose of this study is to extend this methodology for solution of multi-objective optimal control problems under the framework of NSGA-II. The applicability of the technique is illustrated by solving two optimal control problems, taken from literature, which have usually been solved by several methods as single-objective dynamic optimization problems. (C) 2004 Elsevier Ltd. All rights reserved.