Search

Improperly efficient solutions in the sense of Geoffrion in linear fractional vector optimization problems with unbounded constraint sets are studied systematically for the first time in this paper. We give two sets of conditions which assure that all the efficient solutions of a given problem are improperly efficient. We also obtain necessary conditions for an efficient solution to be improperly efficient. As a result, we have new sufficient conditions for Geoffrion’s proper efficiency. The obtained results enrich our knowledge on properly efficient solutions in linear fractional vector optimization.

In this article, a maximum principle for finite horizon state-constrained problems is analyzed via parametric examples. These parametric examples resemble typical optimal growth problems in mathema...

This paper focuses on second-order necessary optimality conditions for constrained optimization problems on Banach spaces. For problems in the classical setting, where the objective function is $$C^2$$ -smooth, we show that strengthened second-order necessary optimality conditions are valid if the constraint set is generalized polyhedral convex. For problems in a new setting, where the objective function is just assumed to be $$C^1$$ -smooth and the constraint set is generalized polyhedral convex, we establish sharp second-order necessary optimality conditions based on the Frechet second-order subdifferential of the objective function and the second-order tangent set to the constraint set. Three examples are given to show that the used hypotheses are essential for the new theorems. Our second-order necessary optimality conditions refine and extend several existing results.

We study a class of finite horizon optimal economic growth problems with nonlinear utility functions and linear production functions. By using a maximum principle in the optimal control theory and employing the special structure of the problems, we are able to explicitly describe the unique solution via input parameters. Economic interpretations of the obtained results and an open problem about the case where the total factor productivity falls into a bounded open interval defined by the growth rate of labor force, the real interest rate, and the exponent of the utility function are also expressed.

The present paper shows how the linear independence constraint qualification (LICQ) can be combined with some conditions put on the first-order and second-order derivatives of the objective function and the constraint functions to ensure the Robinson stability and the Lipschitz-like property of the stationary point set map of a general C2-smooth parametric constrained optimization problem. So, a part of the results in two preceding papers of the authors [J. Optim. Theory Appl. 180 (2019), 91–116 (Part 1); 117–139 (Part 2)], which were obtained for a problem with just one inequality constraint, now has an adequate extension for problems having finitely many equality and inequality constraints. Our main tool is an estimate of B. S. Mordukhovich and R. T. Rockafellar [SIAM J. Optim. 22 (2012), 953–986; Theorem 3.3] for a second-order partial subdifferential of a composite function. The obtained results are illustrated by three examples.

Choo (Oper Res 32:216–220, 1984) has proved that any efficient solution of a linear fractional vector optimization problem with a bounded constraint set is properly efficient in the sense of Geoffrion. This paper studies Geoffrion’s properness of the efficient solutions of linear fractional vector optimization problems with unbounded constraint sets. By examples, we show that there exist linear fractional vector optimization problems with the efficient solution set being a proper subset of the unbounded constraint set, which have improperly efficient solutions. Then, we establish verifiable sufficient conditions for an efficient solution of a linear fractional vector optimization to be a Geoffrion properly efficient solution by using the recession cone of the constraint set. For bicriteria problems, it is enough to employ a system of two regularity conditions. If the number of criteria exceeds two, a third regularity condition must be added to the system. The obtained results complement the above-mentioned remarkable theorem of Choo and are analyzed through several interesting examples, including those given by Hoa et al. (J Ind Manag Optim 1:477–486, 2005).

A series of basic qualitative properties of the minimum sum-of-squares clustering problem are established in this paper. Among other things, we clarify the solution existence, properties of the global solutions, characteristic properties of the local solutions, locally Lipschitz property of the optimal value function, locally upper Lipschitz property of the global solution map, and the Aubin property of the local solution map. We prove that the problem in question always has a global solution and, under a mild condition, the global solution set is finite and the components of each global solution can be computed by an explicit formula. Based on a newly introduced concept of nontrivial local solution, we get necessary conditions for a system of centroids to be a nontrivial local solution. Interestingly, we are able to show that these necessary conditions are also sufficient ones. Finally, it is proved that the optimal value function is locally Lipschitz, the global solution map is locally upper Lipschitz, and the local solution map has the Aubin property, provided that the original data points are pairwise distinct. Thanks to the obtained complete characterizations of the nontrivial local solutions, one can understand better the performance of not only the $k$-means algorithm, but also of other solution methods for the minimum sum-of-squares clustering problem.

This paper presents two new theorems on Geoffrion's properly efficient solutions and seven examples illustrating their applications to linear fractional vector optimization problems with unbounded ...