Mathematical programs with equilibrium constraints.

*(English)*Zbl 0898.90006
Cambridge: Cambridge Univ. Press. xxiv, 401 p. (1997).

This monograph deals with a class of constrained optimization problems which is called Mathematical Programs with Equilibrium Constraints (briefly MPECs). An MPEC is an optimization problem in which the essential constraints are defined by a parametric variational inequality or complementarity system. The class of MPECs is an extension of the class of bilevel programs, also known as mathematical programs with optimization constraints. The MPEC is closely related to the economic problem of Stackelberg game.

This monograph consists of six chapters. Chapter 1 defines the MPEC, gives a brief description of several source problems, and presents various equivalent formulations of the equilibrium constraints in MPEC; the chapter concludes with some results of existence of optimal solutions. Chapter 2 presents an extensive theory of exact penalty functions for MPEC, using the theory of error bounds for inequality systems. This chapter ends with a brief discussion of how some formulations of exact penalty functions of MPECs can be employed to obtain first-order optimality conditions; the latter topic and its extensions are treated in full in the next three chapters. In particular, Chapter 3 presents the fundamental first-order optimality (i.e. stationarity) conditions of MPEC; Chapter 4 verifies in detail the hypotheses needed for the first-order conditions; Chapter 5 contains results on second-order optimality conditions. Chapter 6 presents several algorithms for solving MPECs, including an interior point algorithm for MPECs with “monotone” inner problems, a, conceptual iterative descent algorithm based on an implicit programming approach, and a locally superlinear convergent Newton type (sequential quadratic programming) method based on a piecewise programming approach. Some preliminary computational results are reported.

This monograph consists of six chapters. Chapter 1 defines the MPEC, gives a brief description of several source problems, and presents various equivalent formulations of the equilibrium constraints in MPEC; the chapter concludes with some results of existence of optimal solutions. Chapter 2 presents an extensive theory of exact penalty functions for MPEC, using the theory of error bounds for inequality systems. This chapter ends with a brief discussion of how some formulations of exact penalty functions of MPECs can be employed to obtain first-order optimality conditions; the latter topic and its extensions are treated in full in the next three chapters. In particular, Chapter 3 presents the fundamental first-order optimality (i.e. stationarity) conditions of MPEC; Chapter 4 verifies in detail the hypotheses needed for the first-order conditions; Chapter 5 contains results on second-order optimality conditions. Chapter 6 presents several algorithms for solving MPECs, including an interior point algorithm for MPECs with “monotone” inner problems, a, conceptual iterative descent algorithm based on an implicit programming approach, and a locally superlinear convergent Newton type (sequential quadratic programming) method based on a piecewise programming approach. Some preliminary computational results are reported.

Reviewer: J.Guddat (Berlin)

##### MSC:

90-02 | Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

90C30 | Nonlinear programming |

49J40 | Variational inequalities |

91A65 | Hierarchical games (including Stackelberg games) |