Exceptional Family of Elements for a Variational Inequality Problem and its Applications

This paper introduces a new concept of exceptional family of elements (abbreviated, exceptional family) for a finite-dimensional nonlinear variational inequality problem. By using this new concept, we establish a general sufficient condition for the existence of a solution to the problem. Such a condition is used to develop several new existence theorems. Among other things, a sufficient and necessary condition for the solvability of pseudo-monotone variational inequality problem is proved. The notion of coercivity of a function and related classical existence theorems for variational inequality are also generalized. Finally, a solution condition for a class of nonlinear complementarity problems with so-called P _* -mappings is also obtained.     

Completely Entangled State and Simultaneous Eigenstate in a Finite Dimensional Space

It is shown that a completely entangled state belonging to a finite dimensional Hilbert space is equivalent to a simultaneous eigenstate of two unitary operators. These operators are exponentials of sum and difference of two Hermitian operators of a two-mode system that are complementary with each other.     

The eigenvalue complementarity problem

In this paper an eigenvalue complementarity problem (EiCP) is studied, which finds its origins in the solution of a contact problem in mechanics. The EiCP is shown to be equivalent to a Nonlinear Complementarity Problem, a Mathematical Programming Problem with Complementarity Constraints and a Global Optimization Problem. A finite Reformulation–Linearization Technique (Rlt)-based tree search algorithm is introduced for processing the EiCP via the lattermost of these formulations. Computational experience is included to highlight the efficacy of the above formulations and corresponding techniques for the solution of the EiCP.     

Computing integral solutions of complementarity problems

In this paper an algorithm is proposed to find an integral solution of (nonlinear) complementarity problems. The algorithm starts with a nonnegative integral point and generates a unique sequence of adjacent integral simplices of varying dimension. Conditions are stated under which the algorithm terminates with a simplex, one of whose vertices is an integral solution of the complementarity problem under consideration.     

On a class of least-element complementarity problems

The present paper studies the linear complementarity problem of finding vectorsx andy inR ₊ ⁿ such thatc + Dx + y 0,b – x 0 andx ^T (c + Dx + y) = y ^T (b – x) = 0 whereD is aZ-matrix andb > 0. Complementarity problems of this nature arise, for example, from the minimization of certain quadratic functions subject to upper and lower bounds on the variables. Two least-element characterizations of solutions to the above linear complementarity problem are established first. Next, a new and direct method to solve this class of problems, which depends on the idea of least-element solution is presented. Finally, applications and computational experience with its implementation are discussed.Research partially supported by the National Science Foundation Grant MCS 71-03341 A04 and the Air Force Office of Scientific Research Contract F 44620 14 C 0079.     

Computational complexity of LCPs associated with positive definite symmetric matrices

Murty in a recent paper has shown that the computational effort required to solve a linear complementarity problem (LCP), by either of the two well known complementary pivot methods is not bounded above by a polynomial in the size of the problem. In that paper, by constructing a class of LCPs—one of ordern forn 2—he has shown that to solve the problem of ordern, either of the two methods goes through 2 ⁿ pivot steps before termination.However that paper leaves it as an open question to show whether or not the same property holds if the matrix,M, in the LCP is positive definite and symmetric. The class of LCPs in whichM is positive definite and symmetric is of particular interest because of the special structure of the problems, and also because they appear in many practical applications.In this paper, we study the computational growth of each of the two methods to solve the LCP, (q, M), whenM is positive definite and symmetric and obtain similar results.This research is partially supported by Air Force Office of Scientific Research, Air Force Number AFOSR-78-3646. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation thereon.     

Roles of the respective loops at complementarity determining region on the antigen-antibody recognition

For the rational design of antibody, it is important to clarify the characteristics of the interaction between antigen and antibody. In this study, we evaluated a contribution of the respective complementarity determining region (CDR) loops on the antibody recognition of antigen by performing molecular dynamics simulations for 20 kinds of antigen-antibody complexes. Ser and Tyr showed high appearance rates at CDR loops and the sum of averaged appearance rates of Ser and Tyr was about 20⿿30% at all the loops. For example, Ser and Tyr occupied 23.9% at the light chain first loop (L1) and 23.6% at the heavy chain third loop (H3). The direct hydrogen bonds between antigen and antibody were not equally distributed over heavy and light chains. That is, about 70% of the hydrogen bonds were observed at CDRs of the heavy chain and also the direct hydrogen bond with the shortest distance mainly existed at the loops of the heavy chain for all the complexes. It was revealed from the comparison in contribution to the binding free energy among CDR loops that the heavy chain (especially at H2 and H3) had significant influence on the binding between antigen and antibody because three CDR loops of the heavy chain showed the lowest binding free energy (οG_bind) in 19 complexes out of 20. Tyr in heavy chain (especially in H2 and H3) largely contributed to οG_bind whereas Ser hardly contributed to οG_bind even if the number of the direct hydrogen bond with Ser was the fourth largest and also the appearance rate at CDR was the highest among 20 kinds of amino acid residues. The contributions ofTrp and Phe, which bear aromatic ring in the side chain, were often observed in the heavy chain although the energetic contribution of these residues was not so high as Tyr. The present computational analysis suggests that Tyr plays an outstanding role for the antigen-antibody interaction and the CDR loops of the heavy chain is critically important for antibody recognition of antigen.     

Behavioral measures and their correlation with IPM iteration counts on semi-definite programming problems

We study four measures of problem instance behavior that might account for the observed differences in interior-point method (IPM) iterations when these methods are used to solve semidefinite programming (SDP) problem instances: (i) an aggregate geometry measure related to the primal and dual feasible regions (aspect ratios) and norms of the optimal solutions, (ii) the (Renegar-) condition measure C(d) of the data instance, (iii) a measure of the near-absence of strict complementarity of the optimal solution, and (iv) the level of degeneracy of the optimal solution. We compute these measures for the SDPLIB suite problem instances and measure the sample correlation (CORR) between these measures and IPM iteration counts (solved using the software SDPT3) when these measures have finite values. Our conclusions are roughly as follows: the aggregate geometry measure is highly correlated with IPM iterations (CORR = 0.901), and provides a very good explanation of IPM iterations, particularly for problem instances with solutions of small norm and aspect ratio. The condition measure C(d) is also correlated with IPM iterations, but less so than the aggregate geometry measure (CORR = 0.630). The near-absence of strict complementarity is weakly correlated with IPM iterations (CORR = 0.423). The level of degeneracy of the optimal solution is essentially uncorrelated with IPM iterations. This research has been partially supported through the MIT-Singapore Alliance.     

Complementarity Active-Set Algorithm for Mathematical Programming Problems with Equilibrium Constraints

In this paper, an algorithm for solving a mathematical programming problem with complementarity (or equilibrium) constraints (MPEC) is introduced, which uses the active-set methodology while maintaining the complementarity restrictions throughout the procedure. Finite convergence of the algorithm to a strongly stationary point of the MPEC is established under reasonable hypotheses. The algorithm can be easily implemented by adopting any active-set code for nonlinear programming. Computational experience is included to highlight the efficacy of the proposed method in practice.