A Stackelberg Solution to Joint Optimization Problems: A Case Study of Green Design

AbstractDesign of complex engineered systems often involves optimization of multiple competing problems that are supposed to compromise to arrive at equilibrium optima, entailing a joint optimization problem. This paper reveals the leader-follower decision structure inherent in joint optimization problems. A Stackelberg game solution is formulated to model a leader-follower joint optimization problem as a two-level optimization problem between two decision makers, implicating a mathematical program that contains sub-optimization problems as its constraints. A case study of coffee grinder green design demonstrates the potential of Stackelberg solution to joint optimization of modularity subject with conflicting goals

Monte Carlo and quasi-Monte Carlo sampling methods for a class of stochastic mathematical programs with equilibrium constraints

In this paper, we consider a class of stochastic mathematical programs with equilibrium constraints introduced by Birbil et al. (Math Oper Res 31:739–760, 2006). Firstly, by means of a Monte Carlo method, we obtain a nonsmooth discrete approximation of the original problem. Then, we propose a smoothing method together with a penalty technique to get a standard nonlinear programming problem. Some convergence results are established. Moreover, since quasi-Monte Carlo methods are generally faster than Monte Carlo methods, we discuss a quasi-Monte Carlo sampling approach as well. Furthermore, we give an example in economics to illustrate the model and show some numerical results with this exampl

Monte Carlo and Quasi-Monte Carlo Sampling Methods for a Class of Stochastic Mathematical Programs with Equilibrium Constraints ∗

Abstract. In this paper, we consider a class of stochastic mathematical programs with equilibrium constraints introduced by Birbil et al. (2004). Firstly, by means of a Monte Carlo method, we obtain a nonsmooth discrete approximation of the original problem. Then, we propose a smoothing method together with a penalty technique to get a standard nonlinear programming problem. Some convergence results are established. Moreover, since quasi-Monte Carlo methods are generally faster than Monte Carlo methods, we discuss a quasi-Monte Carlo sampling approach as well. Furthermore, we give an example in economics to illustrate the model and show some numerical results with this example. Key words. Stochastic mathematical program with equilibrium constraints, Monte Carlo/ quasi-Monte Carlo methods, penalization. 2000 Mathematics Subject Classification. 90C30, 90C33, 90C15