Igor Litvinchev

Large-Scale Public R&D Portfolio Selection by Maximizing a Biobjective Impact Measure

This paper addresses R&D portfolio selection in social institutions, state-owned enterprises, and other nonprofit organizations which periodically launch a call for proposals and distribute funds among accepted projects. A nonlinear discontinuous bicriterion optimization model is developed in order to find a compromise between a portfolio quality measure and the number of projects selected for funding. This model is then transformed into a linear mixed-integer formulation to present the Pareto front. Numerical experiments with up to 25 000 projects competing for funding demonstrate a high computational efficiency of the proposed approach. The acceptance/rejection rules are obtained for a portfolio using the rough set methodology.

Aggregation in Large-Scale Optimization

When analyzing systems with a large number of parameters, the dimen- sion of the original system may present insurmountable difficulties for the analysis. It may then be convenient to reformulate the original system in terms of substantially fewer aggregated variables, or macrovariables. In other words, an original system with an n-dimensional vector of states is reformulated as a system with a vector of dimension much less than n. The aggregated variables are either readily defined and processed, or the aggregated system may be considered as an approximate model for the orig- inal system. In the latter case, the operation of the original system can be exhaustively analyzed within the framework of the aggregated model, and one faces the problems of defining the rules for introducing macrovariables, specifying loss of information and accuracy, recovering original variables from aggregates, etc. We consider also in detail the so-called iterative aggregation approach. It constructs an iterative process, at* every step of which a macroproblem is solved that is simpler than the original problem because of its lower dimension. Aggregation weights are then updated, and the procedure passes to the next step. Macrovariables are commonly used in coordinating problems of hierarchical optimization.

A Lagrangian bound for many-to-many assignment problems

A simple procedure to tighten the Lagrangian bounds is proposed. The approach is interpreted in two ways. First, it can be seen as a reformulation of the original problem aimed to split the resulting Lagrangian problem into two subproblems. Second, it can be considered as a search for a tighter estimation of the penalty term arising in the Lagrangian problem. The new bounds are illustrated by a small example and studied numerically for a class of the generalized assignment problems.

A milp bi-objective model for static portfolio selection of R&D projects with synergies

This paper presents a multi-objective MILP model for portfolio selection of research and development (R&D) projects with synergies. The proposed model incorporates information about the funds assigned to different activities as well as about synergies between projects at the activity and project level. The latter aspects are predominant in the context of portfolio selection of R&D projects in public organizations. Previous works on portfolio selection of R&D projects considered interdependencies mainly at the project level. In a few works considering activity level information the models and solution techniques were restricted to problems with a few projects. We study a generalization of our previous model and show that incorporating interdependencies and activity funding information is useful for obtaining portfolios with better quality. Numerical results are presented to demonstrate the efficiency of the proposed approach for large models.

Integer Programming Formulations for Approximate Packing Circles in a Rectangular Container

A problem of packing a limited number of unequal circles in a fixed size rectangular container is considered. The aim is to maximize the (weighted) number of circles placed into the container or minimize the waste. This problem has numerous applications in logistics, including production and packing for the textile, apparel, naval, automobile, aerospace, and food industries. Frequently the problem is formulated as a nonconvex continuous optimization problem which is solved by heuristic techniques combined with local search procedures. New formulations are proposed for approximate solution of packing problem. The container is approximated by a regular grid and the nodes of the grid are considered as potential positions for assigning centers of the circles. The packing problem is then stated as a large scale linear 0-1 optimization problem. The binary variables represent the assignment of centers to the nodes of the grid. Nesting circles inside one another is also considered. The resulting binary problem is then solved by commercial software. Numerical results are presented to demonstrate the efficiency of the proposed approach and compared with known results.

Lagrangian Bounds and a Heuristic for the Two-Stage Capacitated Facility Location Problem

In the two-stage capacitated facility location problem, a single product is produced at some plants in order to satisfy customer demands. The product is transported from these plants to some depots and then to the customers. The capacities of the plants and depots are limited. The aim is to select cost minimizing locations from a set of potential plants and depots. This cost includes fixed cost associated with opening plants and depots, and variable cost associated with both transportation stages. In this work, two different mixed integer linear programming formulations are considered for the problem. Several Lagrangian relaxations are analyzed and compared, and a Lagrangian heuristic producing feasible solutions is presented. The results of a computational study are reported.

Multiperiod optimal planning of thermal generation using cross decomposition

This work addresses the Multiperiod Optimal Planning of Thermal Generation (MOPTG). The model considered is based on a Unit Commitment Problem that has multiperiod character and determines the start up and shut down schedules of thermal plants considering the line capacity limits of transmission and line losses. The mathematical model is stated in the form of a Mixed Integer Non Linear Problem (MINLP) with binary variables. To reduce the computational time caused by the large number of time periods and electric generation nodes we apply the Generalized Cross Decomposition [1, 2]. The later exploits the structure of the problem to reduce solution time by decomposing the MOPTG into a primal subproblem, which is a Non Linear Problem (NLP), a dual subproblem, which is a MINLP, and a Mixed Integer Problem (MIP) called master problem. The approach is compared with Lagrangean Relaxation [3] and Generalized Benders Decomposition [4], To demonstrate the efficiency of the proposed decomposition strategy we present numerical results obtained for three test systems. The computational experiments show the superiority of the Cross Decomposition approach.

Refinement of Lagrangian Bounds in Optimization Problems

Lagrangian constraint relaxation and the corresponding bounds for the optimal value of an original optimization problem are examined. Techniques for the refinement of the classical Lagrangian bounds are investigated in the case where the complementary slackness conditions are not fulfilled because either the original formulation is nonconvex or the Lagrange multipliers are nonoptimal. Examples are given of integer and convex problems for which the modified bounds improve the classical Lagrangian bounds.

An interactive algorithm for portfolio bi-criteria optimization of R&D projects in public organizations

In this paper we propose an interactive algorithm for the selection of portfolios of research and development (R&D) projects in public organizations based on a bi-criteria optimization model, the need for such a model arises when the decision maker (DM) does not trust enough on the portfolio quality measure. This algorithm efficiently exploits the structure and nature of the problem to support the DM. An interesting proposal is also the representation of a portfolio as a set of “rules of support/rejection”; in this way the DM can not only valuate the portfolio by its numerical measures but also compare against his/her beliefs in a way that is more natural for him, which also allows for supporting with more arguments the solution obtained so far. Rough set methodology is employed for rule discovering.

Localization of the optimal solution and posteriori bounds for aggregation

Abstract After an aggregated problem has been solved, it is often desirable to estimate the accuracy loss due to the fact that a simpler problem than the original one has been solved. One way of measuring this loss in accuracy is the difference in objective function values. To get the bounds for this difference, Zipkin (Operations Research 1980;28:406) has assumed, that a simple (knapsack-type) localization of an original optimal solution is known. Since then various extensions of Zipkin’s bound have been proposed, but under the same assumption. A method to compute the bounds for variable aggregation for convex problems, based on general localization of the original solution is proposed. For some classes of the original problem it is shown how to construct the localization. Examples are given to illustrate the main constructions and a small numerical study is presented. Scope and purpose One way of reducing complexity of large-scale optimization problem is aggregation of the problem data and replacing sets of variables by single quantities. Being disaggregated, the solution of an aggregated problem results in a suboptimal solution of the original problem. This paper deals with the determination of error bounds due to aggregation/disaggregation processes. It presents additional ways to reduce these bounds by refinements in the a priori localization region of the optimal solution and by modifying the dual variables of the aggregated solution which appear in the expression of the bounds.