Characterizing optimality in mathematical programming models

Abstract

This paper is a survey of basic results that characterize optimality in single-and multiobjective mathematical programming models. Many people believe, or want to believe, that the underlying behavioural structure of management, economic, and many other systems, generates basically 'continuous' processes. This belief motivates our definition and study of optimality, termed 'structural' optimality. Roughly speaking, we say that a feasible point of a mathematical programming model is structurally optimal if every improvement of the optimal value function, with respect to parameters, results in discontinuity of the. corresponding feasible set of decision variables. This definition appears to be more suitable for many applications and it is also more general than the usual one: every optimum is a structural optimum but not necessarily vice versa. By characterizing structural optima, we obtain some new, and recover the familiar, optimality conditions in nonlinear programming.The paper is self-contained. Our approach is geometric and inductive: we develop intiution by studying finite-dimensional models before moving on to abstract situations.AMS subject clmsilieatiom (1980). 90C25, 90C48, 90C31, 54C10. 54C60, 49B27, 49B50.

Partly convex programming and zermelo's navigation problems

“…Indeed, it is well known (see, e.g., [21,22]) that a local minimum of a function is also global, if the feasible set mapping, relative to the right-hand side perturbations, is lower semi-continuous. The importance of lower semicontinuity in characterizations of local optima in parametric optimization has been demonstrated in, e.g., [25]. This paper reconfirms that, in order to characterize global and local optima in nonconvex optimization, in addition to linear algebra and calculus, one also needs some basic tools from point-to-set topology.…”

“…Also (see, e.g., [25]) F is lower semicontinuous at 0 h. Moreover, the KKT conditions are satisfied at x k, the Lagrangian becomes…”

“…Recall that a point-to-set mapping F 9 R p --+ R n is said to be lower semicontinuous at 0* E R p if for every open set ,A c R ~ such that .,4 f3 F(0*) # 0 there exists a neighbourhood N(O*) of 0* such that ,4 M F(0) # 0 for every 0 E N(O*). (See, e.g., [3,25].) (ii) For the usual convex programs (the case z = x, 0 = r in PCP), S(O*) = implying that every feasible point is globally cooperative.…”

“…(i) It is well known (see [25,26]) that for lower semicontinuous pointto-set feasible set mapping at the 0 level: F 9 0 --+ F(#), the set S(0*) contains a neighbourhood of 0". Therefore, for such programs, every feasible point is at least locally cooperative.…”

“…We recall (see [25,Theorem 2.2]) that under the assumption that the set of optimal solutions of P(O k) is nonempty and bounded and the mapping F is lower semicontinuous at O k relative to 9 c, there where x kt = x kl (0 ~t) is an optimal solution of the program P(Okt). (For conditions that guarantee the existence of these limits see, e.g., [16].…”