The continuity of the optimum in parametric programming and applications to stochastic programming

It is proved a sufficient condition that the optimal value of a linear program be a continuous function of the coefficients. The condition isessential, in the sense that, if it is not imposed, then examples with discontinuous optimal-value function may be found. It is shown that certain classes of linear programs important in applications satisfy this condition. Using the relation between parametric linear programming and the distribution problem in stochastic programming, a necessary and sufficient condition is given that such a program has optimal value. Stable stochastic linear programs are introduced, and a sufficient condition of such stability, important in computation problems, is established.This note is a slightly modified version of a paper presented at the Institute of Econometrics and Operations Research of the University of Bonn, Bonn, Germany, 1972.The author is grateful to G. B. Dantzig and S. Karamardian for useful comments on an earlier draft of this paper. In particular, S. Karamardian proposed modifications which made clearer the proof of Lemma 2.1.     

The range of the efficient frontier in multiple objective linear programming

Because a rational decision maker should only select an efficient alternative in multiple criterion decision problems, the efficient frontier defined as the set of all efficient alternatives has become a central solution concept in multiple objective linear programming. Normally this set reduces the set of available alternatives of the underlying problem. There are several methods, mainly based on the simplex method, for computing the efficient frontier. This paper presents a quite different approach which uses a nonlinear parametric program, solved by Wolfe's algorithm, to determine the range of the efficient frontier.     

Directional derivatives of the solution of a parametric nonlinear program

Consider a parametric nonlinear optimization problem subject to equality and inequality constraints. Conditions under which a locally optimal solution exists and depends in a continuous way on the parameter are well known. We show, under the additional assumption of constant rank of the active constraint gradients, that the optimal solution is actually piecewise smooth, hence B-differentiable. We show, for the first time to our knowledge, a practical application of quadratic programming to calculate the directional derivative in the case when the optimal multipliers are not unique.This author's research was supported by the Australian Research Council.     

利用自发参量下转换产生795nm关联光子对的实验研究

本文进行了自发参量下转换(SPDC)产生795nm关联光子对的实验研究。选择Ⅱ类PPKTP晶体作为非线性介质,用倍频获得的397.5nm紫光作为泵浦光。在实验上首先通过符合计数测量了紫光单次穿过PPKTP晶体时SPDC过程产生的光子对亮度,当泵浦光功率为5mW时,测得符合计数为53 150/s。然后将下转换过程放入光学谐振腔中,对下转换光子的线宽进行压窄。实验上测得下转换光子对的关联时间为14.6ns,对应线宽约为15MHz。本文为纠缠光子对在Rb原子系综中的存储提供了实验基础。     

Editorial Policy

Consider the parametric linear complementarity problem w=Mz+q+p, w0, z0, w ^T z=0, where p0, 0q0, and 0. We show that a necessary condition for every complementary map z() to be isotone for every nonzero q0 and every p is that M be either a P-matrix or a -matrix. The Cottle necessary and sufficient conditions for strong and uniform isotonicity for P-matrices are restated, with slight modifications, for -matrices.     

Stability and sensitivity of solutions to nonlinear optimal control problems

Parameter-dependent optimal control problems for nonlinear ordinary differential equations, subject to control and state constraints, are considered. Sufficient conditions are formulated under which the solutions and the associated Lagrange multipliers are locally Lipschitz continuous and directionally differentiable functions of the parameter. The directional derivatives are characterized.This research was partially supported by Grant No. 3 0256 91 01 from Komitet Bada Naukowych.     

On structure and stability in stochastic programs with random technology matrix and complete integer recourse

For two-stage stochastic programs with integrality constraints in the second stage, we study continuity properties of the expected recourse as a function both of the first-stage policy and the integrating probability measure.Sufficient conditions for lower semicontinuity, continuity and Lipschitz continuity with respect to the first-stage policy are presented. Furthermore, joint continuity in the policy and the probability measure is established. This leads to conclusions on the stability of optimal values and optimal solutions to the two-stage stochastic program when subjecting the underlying probability measure to perturbations.This research is supported by the Schwerpunktprogramm Anwendungsbezogene Optimierung und Steuerung of the Deutsche Forschungsgemeinschaft.The main part of the paper was written while the author was an assistant at the Department of Mathematics at Humboldt University Berlin.     

On unpredictable ship rolling in irregular seas

Indirectly excited large amplitude roll motions of container ships and roll-on/roll-off ferries represent a considerable threat to goods and life in modern navigation. Especially when ships are heading into rough seas, or sailing away from following seas, large roll responses have been reported to occur in an unpredictable, random way. We approach this topic from deterministic bifurcation theory and generalize the analysis by characterizing the seaway as narrow band stochastic processes. It is demonstrated how the intermittent behavior of zero and large amplitude motions in irregular seas is related to the deterministic bifurcation scenario. Further, we identify parameter regions of intermittent roll motions and show how the on-/off intermittency can be characterized by the probability density function of the response amplitude.