Narrow Results

Multilevel programming appears in many decision-making situations. Investigation of the main properties of quasiconcave multilevel programming (QCMP) problems, to date, is limited to bilevel programming (only two levels). In this paper, first, we present an extension of the properties of quasiconcave bilevel programming (QCBP) problems for the case when three levels exist. Then, by induction on n (the number of levels), we prove the existence of an extreme point of the polyhedral constraint region that solves the QCMP problem under given conditions. Ultimately, a number of numerical examples are illustrated to verify the results.

Interval programming is one of main approaches treating imprecise and uncertain elements involved in optimization problems. In this paper, an interval linear fractional bilevel program is considered, which is characterized in that both objective coefficients and the right-hand side vector are interval numbers, and an evolutionary algorithm (EA) is proposed to solve the problem. First, the interval parameter space of the follower’s problem is taken as the search space of the proposed EA. For each individual, one can evaluate it by dealing with a simplified interval bilevel program in which only the leader’s objective involves interval parameters. In addition, the optimality conditions of linear fractional programs are applied to convert and solve the simplified problem. Finally, some computational examples were solved and the results show that the proposed algorithm is efficient and robust.

A new systematic approach to holonomic constraint is presented. By holonomic we mean a constraint problem in the calculus of variations/optimal control theory where the constraints are independent of the derivative of the dependent variable. It is seen that these new methods follow from a general theory of constraint optimization previously given by the author. A major, new emphasis of this work is the necessity of properly handling the boundary values of our introduced variables.

The author's previous theory allows the solution of a wide variety of general or anholonomic problems which include those identified with the areas of control, delay, stochastic, and partial differential equations in the sense that simpler constraint problems in these areas can be solved exactly by analytical means while more complex problems can be solved by efficient numerical algorithms with an a priori maximal error of O(h²), where h is the node size. Thus, our methods in this paper are immediately applicable to holonomic problems in these other diverse areas.

While our new results do not involve a significant formal mathematical jump from the author's previous theory of constraint optimization, the treatment of boundary values is significant and of note. For this reason we will give several examples to illustrate these ideas. Finally, we illustrate how our methods can be used to solve a wide variety of meaningful holonomic constraint or generalized coordinate problems in the study of dynamics/classical mechanics, using the simple pendulum problem as an example.