New Direction to the Scheduling Problem

This chapter presents a new direction to the scheduling problem by exploring the Moore-Hodgson algorithm. This algorithm is used within the context of integer programming to come up with complementarity conditions, more biding constraints, and a strong lower bound for the scheduling problem. With Moore-Hodgson Algorithm, the alternate optimal solutions cannot be easily generated from one optimal solution; however, with integer formulation, this is not a problem. Unfortunately, integer formulations are sometimes very difficult to handle as the number jobs increases. Therefore, the integer formulation presented in this chapter uses infeasibility to verify optimality with branch and bound related algorithms. Thus, the lower bound was obtained using pre-processing and shown to be highly accurate and on its own can be used in those situations where quick scheduling decisions are required.

Complementarity Relationships and Critical Configurations in Rigid-Body Collisions of Planar Kinematic Chains With Smooth External Contacts

In this article, we consider a special class of collision problems that are frequently encountered in the field of robotics. Such problems can be described as a kinematic chain with one of its ends striking an external surface, while the remaining ends resting on other surfaces. This type of problem involves complementarity relationships between the normal velocities and impulses at the contacting ends. We present a solution method that takes into account the complementarity conditions at the contacting ends. In addition, we study the critical configurations of particle and rigid-body chains where the impulse wave generated by impact gets blocked before it reaches a contacting end.

Small deformation rate-independent plasticity based on a postulate of maximum dissipation

This chapter introduces the concept of maximum dissipation. The elastic set is introduced, and the plastic dissipation is maximized over the elastic set using classical methods from linear programming theory. The plastic flow direction is seen to be generally normal to the yield surface when the plastic dissipation is maximized. The Kuhn-Tucker complementarity conditions are seen in this context to arise from the postulated optimization problem, and the elastic set is seen to be necessarily convex. The concept of maximum dissipation is applied to a Mises material and the models of the earlier chapters are seen to be recovered.

2-factor Newton method for solving the constrained optimization problem with the singular Kuhn—Tucker system

A new method for solving the inequality constrained optimization problem is proposed for the case when the system of necessary optimality conditions of Kuhn—Tucker is degenerate. This situation occurs for example in the case when strict complementarity conditions fails in solution point. The reduction of the inequalities con- strained optimization problem to the equalities constrained problem is substantiated and the use of a new 2-fac- tor Newton method for the effective solution of the obtained degenerate system of optimality conditions is shown.

A new smoothing method for solving nonlinear complementarity problems

In this paper, a new improved smoothing Newton algorithm for the nonlinear complementarity problem was proposed. This method has two-fold advantages. First, compared with the classical smoothing Newton method, our proposed method needn’t nonsingular of the smoothing approximation function; second, the method also inherits the advantage of the classical smoothing Newton method, it only needs to solve one linear system of equations at each iteration. Without the need of strict complementarity conditions and the assumption of P0 property, we get the global and local quadratic convergence properties of the proposed method. Numerical experiments show that the efficiency of the proposed method.

Study of compositional multiphase flow formulation using complementarity conditions

In this article, two formulations of multiphase compositional Darcy flows taking into account phase transitions are compared. The first formulation is the so-called natural variable formulation commonly used in reservoir simulation, the second has been introduced by Lauseret al.and uses the phase pressures, saturations and component fugacities as main unknowns. We will discuss how the Coats and the Lauser approaches can be used to solve a compositional multiphase flow problem with cubic equations of state of Peng and Robinson. Then, we will study the results of several synthetic cases that are representative of petroleum reservoir engineering problems and we will compare their numerical behavior.