On Direct Methods for Lexicographic Min-Max Optimization

The approach called the Lexicographic Min-Max (LMM) optimization depends on searching for solutions minimal according to the lex-max order on a multidimensional outcome space. LMM is a refinement of the standard Min-Max optimization, but in the former, in addition to the largest outcome, we minimize also the second largest outcome (provided that the largest one remains as small as possible), minimize the third largest (provided that the two largest remain as small as possible), and so on. The necessity of point-wise ordering of outcomes within the lexicographic optimization scheme causes that the LMM problem is hard to implement. For convex problems it is possible to use iterative algorithms solving a sequence of properly defined Min-Max problems by eliminating some blocked outcomes. In general, it may not exist any blocked outcome thus disabling possibility of iterative Min-Max processing. In this paper we analyze two alternative optimization models allowing to form lexicographic sequential procedures for various nonconvex (possibly discrete) LMM problems. Both the approaches are based on sequential optimization of directly defined artificial criteria. The criteria can be introduced into the original model with some auxiliary variables and linear inequalities thus the methods are easily implementable. 1 Lexicographic Min-Max There are several multiple criteria decision problems where the Pareto-optimal solution concept is not powerful enough to resolve the problem since the equity or fairness among uniform individual outcomes is an important issue [10, 11, 17]. Uniform and equitable outcomes arise in many dynamic programs where individual objective functions represent the same outcome for various periods [9]. In the stochastic problems uniform objectives may represent various possible values of the same nondeterministic outcome ([15] and references therein). Moreover, many modeling techniques for decision problems first introduce some uniform objectives and next consider their impartial aggregations. The most direct models with uniform equitable criteria are related to the optimization of systems The research was supported by the Ministry of Science and Information Society Technologies under grant 3T11C 005 27 “Models and Algorithms for Efficient and Fair Resource Allocation in Complex Systems.” M. Gavrilova et al. (Eds.): ICCSA 2006, LNCS 3982, pp. 802–811, 2006. c © Springer-Verlag Berlin Heidelberg 2006 On Direct Methods for Lexicographic Min-Max Optimization 803 which serve many users. For instance, efficient and fair way of distribution of network resources among competing demands becomes a key issue in computer networks [5] and the telecommunication networks design, in general [20]. The generic decision problem we consider may be stated as follows. There is given a set I of m clients (users, services). There is also given a set Q of feasible decisions. For each service j ∈ I a function fj(x) of the decision x is defined. This function, called the individual objective function, measures the outcome (effect) yj = fj(x) of the decision for client j. An outcome can be measured (modeled) as service time, service costs, service delays as well as in a more subjective way as individual utility (or disutility) level. In typical formulations a smaller value of the outcome means a better effect (higher service quality or client satisfaction). Therefore, without loss of generality, we can assume that each individual outcome yi is to be minimized. The Min-Max solution concept depends on optimization of the worst outcome min x { max j=1,...,m fj(x) : x ∈ Q } and it is regarded as maintaining equity. Indeed, for a simplified resource allocation problem min{maxj yj : ∑ j yj ≤ b } the Min-Max solution takes the form ȳj = b/m for all j ∈ I thus meeting the perfect equity. In the general case with possibly more complex feasible set structure this property is not fulfilled. Actually, the distribution of outcomes may make the Min-Max criterion partially passive when one specific outcome is relatively large for all the solutions. For instance, while allocating clients to service facilities, such a situation may be caused by existence of an isolated client located at a considerable distance from all facilities. Minimization of the maximum distance is then reduced to that single isolated client leaving other allocation decisions unoptimized. This is a clear case of inefficient solution where one may still improve other outcomes while maintaining fairness (equitability) by leaving at its best possible value the worst outcome. The Min-Max solution may be lexicographically regularized according to the Rawlsian principle of justice [22]. Applying the Rawlsian approach, any two states should be ranked according to the accessibility levels of the least well–off individuals in those states; if the comparison yields a tie, the accessibility levels of the next–least well–off individuals should be considered, and so on. Formalization of this concept leads us to the lexicographic Min-Max optimization. Let 〈a〉 = (a〈1〉, a〈2〉, . . . , a〈m〉) denote the vector obtained from a by rearranging its components in the non-increasing order. That means a〈1〉 ≥ a〈2〉 ≥ . . . ≥ a〈m〉 and there exists a permutation π of set I such that a〈i〉 = aπ(i) for i ∈ I. Comparing lexicographically such ordered vectors 〈y〉 one gets the so-called lex-max order. The general problem we consider depends on searching for the solutions that are minimal according to the lex-max order: lexmin x {(θ1(f(x)), . . . , θm(f(x))) : x ∈ Q} where θj(y) = y〈j〉 (1) The lexicographic Min-Max under consideration is related to the problems with outcomes being minimized. Similar consideration of the maximization problems 804 W. Ogryczak and T. Śliwiński leads to the lexicographic Max-Min solution concept. Obviously, all the results presented further for the lexicographic Min-Max can be adjusted to the lexicographic Max-Min while preserving assumption that the outcomes are ordered from the worst one to the best one. The lexicographic Min-Max solution is known in game theory as the nucleolus of a matrix game. It originates from an idea [6] to select from the optimal strategy set those which allow one to exploit mistakes of the opponent optimally. It has been later refined to the formal nucleolus definition [21]. The concept was early considered in the Tschebyscheff approximation [23] as a refinement taking into account the second largest deviation, the third one and further to be hierarchically minimized. Similar refinement of the fuzzy set operations has been recently analyzed [7]. Within the telecommunications or network applications the lexicographic Max-Min approach has appeared already in [2] and now under the name Max-Min Fairness (MMF) is treated as one of the standard fairness concepts [16, 20]. The LMM approach has been used for general linear programming multiple criteria problems [1, 12], as well as for specialized problems related to (multiperiod) resource allocation [9, 11]. Note that the lexicographic minimization in the LMM is not applied to any specific order of the original criteria. Nevertheless, in the case of linear programming (LP) problems (or generally convex optimization), there exists a dominating objective function which is constant (blocked) on the entire optimal set of the Min-Max problem [12]. Hence, having solved the Min-Max problem, one may try to identify the blocked objective and eliminate it to formulate a new restricted Min-Max problem on the former optimal set. Therefore, the LMM solution to LP problems can be found by the sequential Min-Max optimization with elimination of the blocked outcomes. The LMM approach has been considered also for various discrete optimization problems [3, 4, 8] including the location-allocation ones [14]. In discrete models, due to the lack of convexity there may not exist any blocked outcome [13] thus disabling possibility of the sequential Min-Max algorithm. In this paper we analyze capabilities of an effective use of earlier developed ordered cumulated outcomes methodology [17, 18, 19] to solve the LMM problem by sequential optimization of directly defined criteria. We develop and analyze two alternative approaches allowing to form lexicographic sequential procedures for various nonconvex (possibly discrete) LMM problems. Both the approaches are based on criteria directly introduced with some LP expansion of the original model.