On lexicographic optimal solutions in transportation problems

Lexicographic versions of the cost minimizing transportation problem (CMTP) and the time minimizing transportation problem (TMTP) are presented in this paper. In addition to minimizing the quantity sent on the costliest routes in a cost minimizing transportation problem. an attempt is made to minimize the quantity transported on the second-costliest routes. if the shipment on the costliest routes is as small as possible and the quantity shipped on the third-costliest routes, if the shipments on the costliest and the second- costliest routes are as small as possible. and so on. In a lexicographic time minimizing transportation problem one is not only interested in minimizing the transportation cost on the routes of the longest duration but also on the routes of second longest, third-longest duration and so on. For finding lexicographic optimal solutions (LOS) of lexicographic cost minimizing and time minimizing transportation problems a standard cost minimizing transportation problem is formulated whose optimal solution is shown to provide the answer. Some extensions are also discussed     

A PDE approach to regularity of solutions to finite horizon optimal switching problems

We study optimal 2-switching and n-switching problems and the corresponding system of variational inequalities. We obtain results on the existence of viscosity solutions for the 2-switching problem for various setups when the cost of switching is non-deterministic. For the n-switching problem we obtain regularity results for the solutions of the variational inequalities. The solutions are C^1,1-regular away for the free boundaries of the action sets.     

On the existence of sporadically catching-up optimal solutions for infinite-horizon optimal control problems

In this paper, we extend the existence theory of Brock and Haurie concerning the existence of sporadically catching-up optimal solutions for autonomous, infinite-horizon optimal control problems. This notion of optimality is one of a hierarchy of types of optimality that have appeared in the literature to deal with optimal control problems whose cost functionals, described by an improper integral, either diverge or are unbounded below. Our results rely on the now classical convexity and seminormality hypotheses due to Cesari and are weaker than those assumed in the work of Brock and Haurie. An example is presented where our results are applicable, but those of the above-mentioned authors do not.This research forms part of the author's doctoral dissertation, written at the University of Delaware, Newark, Delaware, under the supervision of Professor T. S. Angell.     

Minimal regularity of the solutions of some transmission problems

We consider some transmission problems for the Laplace operator in two‐dimensional domains. Our goal is to give minimal regularity of the solutions, better than H¹, with or without conditions on the (positive) material constants. Under a monotonicity or quasi‐monotonicity condition on the constants (or on the inverses according to the boundary conditions), we study the behaviour of the solution near vertex and near interior nodes and show in each case that the given regularity is sharp. Without condition we prove that the regularity near a corner is of the form H^1+ρ, where ρ is a given bound depending on the material constants. Numerical examples are presented which confirm the sharpness of our lower bounds. Copyright © 2003 John Wiley & Sons, Ltd.     

障碍问题解的内正则性

对非幂次增长的障碍问题 :∫Ωai(x,u,Du) φ xidx + ∫Ωb(x,u,Du)φ dx≥ 0　　这里φ(x)≥ψ(x) - u(x) ,u(x)≥ψ(x) ,而φ∈ W1 0 LM(Ω ) ,ψ为局部 Holder连续的 ,我们得到其在 W1 LM(Ω)中弱解的 C0 ,αloc 正则性     

On the existence of optimal solutions to integer and mixed-integer programming problems

The purpose of this paper is to present sufficient conditions for the existence of optimal solutions to integer and mixed-integer programming problems in the absence of upper bounds on the integer variables. It is shown that (in addition to feasibility and boundedness of the objective function) (1) in the pure integer case a sufficient condition is that all of the constraints (other than non-negativity and integrality of the variables) beequalities, and (2) that in the mixed-integer caserationality of the constraint coefficients is sufficient. Some computational implications of these results are also given.     

On the dependence of the solutions and optimal solutions of control problems on the control constraint set

In this paper we examine the dependence of the solutions and optimal solutions of a class of linear, infinite dimensional control systems on the control constraint set. This is done using the weak and the Kuratowski-Mosco convergence of sets. First we establish some general facts about weakly convergent multifunctions. Then we prove some convergence theorems for the trajectories of certain control systems. We also derive a general relaxation theorem. Subsequently we pass to optimal control problems and prove various convergence results. We conclude with an example from parabolic control systems.Research supported by N. S. F. Grant DMS-8802688     

On proper and improper efficient solutions of optimal problems with multicriteria

The properness of the efficient solution of the optimal problem with multicriteria has been independently defined by Kuhn and Tucker, Geoffrion, and Klinger. A theorem of Geoffrion describes the relation between Geoffrion's and Kuhn and Tucker's properness. In this paper, the dual part of the theorem is given, and some geometric approach is applied to derive the optimal conditions of proper efficient solutions and improper efficient solutions.     

Application of classical transportation methods to find the fuzzy optimal solution of fuzzy transportation problems

To the best of our knowledge till now there is no method in the literature to find the exact fuzzy optimal solution of unbalanced fully fuzzy transportation problems. In this paper, the shortcomings and limitations of some of the existing methods for solving the problems are pointed out and to overcome these shortcomings and limitations, two new methods are proposed to find the exact fuzzy optimal solution of unbalanced fuzzy transportation problems by representing all the parameters as LR flat fuzzy numbers. To show the advantages of the proposed methods over existing methods, a fully fuzzy transportation problem which may not be solved by using any of the existing methods, is solved by using the proposed methods and by comparing the results, obtained by using the existing methods and proposed methods. It is shown that it is better to use proposed methods as compared to existing methods.     

Existence of finitely optimal solutions for infinite-horizon optimal control problems

In this paper, we investigate the existence of finitely optimal solutions for the Lagrange problem of optimal control defined on [0, ) under weaker convexity and seminormality hypotheses than those of previous authors. The notion of finite optimality has been introduced into the literature as the weakest of a hierarchy of types of optimality that have been defined to permit the study of Lagrange problems, arising in mathematical economics, whose cost functions either diverge or are not bounded below. Our method of proof requires us to analyze the continuous dependence of finite-interval Lagrange problems with respect to a prescribed terminal condition. Once this is done, we show that a finitely optimal solution can be obtained as the limit of a sequence of solutions to a sequence of corresponding finite-horizon optimal control problems. Our results utilize the convexity and seminormality hypotheses which are now classical in the existence theory of optimal control.This research forms part of the author's doctoral dissertation written at the University of Delaware, Newark, Delaware under the supervision of Professor Thomas S. Angell.