Yo Ishizuka

Sensitivity and stability analysis for nonlinear programming

We give a brief overview of important results in several areas of sensitivity and stability analysis for nonlinear programming, focusing initially on “qualitative” characterizations (e.g., continuity, differentiability and convexity) of the optimal value function. Subsequent results concern “quantitative” measures, in particular optimal value and solution point parameter derivative calculations, algorithmic approximations, and bounds. Our treatment is far from exhaustive and concentrates on results that hold for smooth well-structured problems.

Double penalty method for bilevel optimization problems

A penalty function method approach for solving a constrained bilevel optimization problem is proposed. In the algorithm, both the upper level and the lower level problems are approximated by minimization problems of augmented objective functions. A convergence theorem is presented. The method is applicable to the non-singleton lower-level reaction set case. Constraint qualifications which imply the assumptions of the general convergence theorem are given.

Mean and variance of waiting time and their optimization for alternating traffic control systems

We analyze alternating traffic crossing a narrow one-lane bridge on a two-lane road. Once a car begins to cross the bridge in one direction, arriving cars from the other direction must wait, forming a queue, until all the arrivals in the first direction finish crossing the bridge. Such a situation can often be observed when road-maintenance work is being carried out. Cars are assumed to arrive at the queues according to independent Poisson processes and to cross the bridge in a constant time. In addition, once cars join the queue, each car needs a constant starting delay, before starting to cross the bridge. We model the situation where a signal controls the traffic so that the signal gives a priority to one direction as long as a new car from the same direction arrives in a fixed time. For this model, we get a closed form for the first two moments of the waiting time of cars arriving at the bridge, and then numerically obtain Pareto optimal solutions of holding times to minimize the mean waiting time and its standard deviation.

Two-Level Mathematical Programming Problem

In the latter half of this book we are concerned with theories and solution methods for various two-level optimization problems, which are generally called two-level mathematical programs. The purpose of this chapter is to present a unified treatment of various two-level optimization problems in the context of general two-level nonlinear programming. In so doing, we show how several variants fit the general model. As such, one can achieve a unified framework for each individual problem discussed in the following chapters.

Two-Level Design Problem (Mathematical Programming with Optimal-Value Functions)

This chapter is concerned with a two-level design problem [S23, T1, I4] in which the central system makes a decision on parameter values to be assigned to the subsystems so as to optimize its objective, taking the values of the subsystem performance into account. In the second stage, the subsystems optimize their individual objective functions under the given parameters from the center. Such a problem is mathematically formulated as an optimization problem whose objective and constraint functions include the optimal-value functions obtained from the optimization of the subsystem performance indices.

Satisfaction Optimization Problem

This chapter deals with an optimization problem involving unknown parameters (uncertainty). We consider a decision problem whose objective function is minimized under the condition that a certain performance function should always be less than or equal to a prescribed permissible level (for every value of the unknown parameters). In the case that the set in which the unknown parameters must lie contains an infinite number of elements, we say that the corresponding optimization problem has an infinite number of inequality constraints and call it an infinitely constrained optimization problem.†

The Stackelberg Problem: Linear and Convex Case

The vast majority of research on two-level programming has centered on the linear Stackelberg game, alternatively known as the linear bilevel programming problem (BLPP). In this chapter we present several of the most successful algorithms that have been developed for this case, and when possible, compare their performance. We begin with some basic notation and a discussion of the theoretical character of the problem.

Large-Scale Nonlinear Programming: Decomposition Methods

The use of primal and dual methods are at the heart of finding solutions to large-scale nonlinear programming problems. Both methods are algorithms of a two-level type where the lower-level decision makers work independently to solve their individual subproblems generated by the decomposition of the original (overall) problem. At the same time, the upper-level decision maker solves his coordinating problem by using the results coming from the lower-level optimizations. These algorithms perform optimization calculations successively by an iterative exchange of information between the two levels.

Min-Max Problem

The min-max problem is a model for decision making under uncertainty. The aim is to minimize the function f (x, y) but the decision maker only has control of the vector x ∈ R n . After he selects a value for x, an “opponent” chooses a value for y ∈ R m which alternatively can be viewed as a vector of disturbances. When the decision maker is risk averse and has no information about how y will be chosen, it is natural for him to assume the worst. In other words, the second decision maker is completely antagonistic and will try to maximize f (x,y) once x is fixed. The corresponding solution is called the min-max solution and is one of several conservative approaches to decision making under uncertainty. When stochastic information is available for y other approaches might be more appropriate (e.g., see [S4, E3]).

The Stackelberg Problem: General Case

The Stackelberg problem is the most challenging two-level structure that we examine in this book. It has numerous interpretations but originally it was proposed as a model for a leader-follower game in which two players try to minimize their individual objective functions F(x, y) and f (x, y), respectively, subject to a series of interdependent constraints [S28, S27]. Play is defined as sequential and the mood as noncooperative. The decision variables are partitioned between the players in such a way that neither can dominate the other. The leader goes first and through his choice of x ∈ R n is able to influence but not control the actions of the follower. This is achieved by reducing the set of feasible choices available to the latter. Subsequently, the follower reacts to the leader’s decision by choosing a y ∈ R m in an effort to minimizes his costs. In so doing, he indirectly affects the leader’s solution space and outcome.