This is the first of three chapters in which we derive some necessary optimality conditions for the MPEC (1.1.1). This chapter is concerned with the fundamental first-order conditions; Chapter 4 deals with the verification of hypotheses required for these first-order conditions; and Chapter 5 is concerned with the second-order conditions. Although we have seen that the MPEC (1.1.1) can be formulated as a standard nonlinear program via the KKT formulation of the inner VIs (see, e.g., Theorem 1.3.5), for reasons to be given in this chapter a straightforward application of the optimality conditions in classical nonlinear programming is inappropriate.

One approach to deriving optimality conditions of an MPEC is to use a differentiable exact penalty equivalent of the problem, such as (2.4.7) or (2.4.10), and apply classical results to such a formulation. A drawback of this approach is that strong assumptions are required for the MPEC to be equivalent to a differentiable constrained optimization problem; see Theorem 2.4.7. Even in the case of the MPAEC (Theorem 2.4.4), the equivalent formulation requires the restrictive nondegeneracy assumption. For the MPAEC, one could use the less restrictive Theorem 2.4.3; although the min function is nondifferentiable, it is simple enough to make the formulation (2.4.6) a viable candidate for this purpose; see Subsection 2.4.3. Proposition 2.4.9 provides a preliminary set of necessary conditions for a local minimum of the MPAEC.

In this chapter, we abandon the exact penalty function approach for the derivation of optimality conditions of MPEC. Instead, we resort to first principles to understand the stationarity conditions and the tangent cone of the feasible region of MPEC. The chapter contains four sections.

As for a standard NLP, we can derive some second-order necessary and second-order sufficient conditions for a local minimum of an MPEC. This chapter is a foray into these second-order conditions. For convenience, throughout we require polyhedrality of the upper-level feasible region Z; initially, we also assume the same for the constraint set C(x) in the lower-level VI for all x ∈ dom(C). In standard nonlinearly constrained NLPs, second-order conditions at boundary points must generally account for the curvature of the boundary. Such curvature requirement is usually contained in the positive definiteness properties of the partial Hessian matrix of the Lagrangean function of the nonlinear program, which contains not only the Hessian matrix of the objective function, but also the sum of the Hessian matrices of the active constraint functions, using the KKT multipliers as weights; cf. the discussion on SCOC in Subsection 4.2.7. Initially we confine our interest to polyhedral sets to keep the ideas and analysis relatively simple. Even in this situation, the treatment is complicated by the equilibrium constraint (x,y) ∈ Gr S which, as we have seen in the case of the first-order analysis, leads to some combinatorial considerations that are not present in standard nonlinear programming. Indeed, such complications become more pronounced as we work with the second-order conditions in this chapter.

This chapter discusses a multiplier-based approach, an implicit programming approach, and a piecewise programming approach to the derivation of second-order optimality conditions of MPECs; the treatment here extends ideas of the previous two chapters. Overall, the development here is parallel to that of Chapter 4. Specifically, besides the next section which reviews known NLP theory, there are five sections in the chapter.

In Section 3.3, we have introduced the full, extreme, and basic CQs for an MPEC. Under any one of these CQs, stationarity of a feasible solution to an MPEC can be characterized; see Corollary 3.3.1, Theorem 3.3.4, and Remark 3.3.5. In this chapter, we consider some important cases of the general MPEC (3.1.1) and introduce certain assumptions in order to verify such CQs, thus completing the task of deriving first-order optimality conditions for the MPEC.

This chapter has four main sections. Section 4.1 considers the AVI constrained mathematical program, i.e., the MPAEC. The main goal of this section is to demonstrate that the extreme CQ must hold without any particular assumptions. This conclusion is consistent with a standard nonlinear program with linear constraints for which the KKT conditions must be necessary for local optimality. The next two sections, 4.2 and 4.3, discuss two general approaches to deal with the MP with nonlinear VI constraints. The approach in Section 4.2 is of interest because of the bilevel nature of the MPEC and the special significance of the variables x and y, whereas the approach in Section 4.3 is more along the line of traditional nonlinear programming, but with an important modification to deal with the combinatorial feature of the MPEC. The chapter concludes with a discussion of an exact penalization of order 1, using the assumptions set forth in Section 4.2.

The two approaches discussed in Sections 4.2 and 4.3 together provide a powerful tool for the study of the MPEC. They are based on different assumptions and are therefore applicable to different classes of MPECs.

In the previous three chapters, we studied the necessary optimality conditions and sufficient optimality conditions of the MPEC. This chapter discusses three general classes of iterative algorithms for computing a stationary point of an MPEC. The first class of algorithms is based on a penalty interior point approach (PIPA); the second class of algorithms is based on an implicit programming approach. The third class of algorithms is of the Newton type for solving MPEC, based on the piecewise programming formulation. For the latter algorithms, some conditions pertaining to the relaxed nonlinear program introduced at the end of Subsection 4.3.1 turn out to provide useful conditions for their convergence. In essence, the interior point approach is applicable to MPECs where the lower-level problems possess certain generalized monotonicity properties; the second approach relies on the implicit program formulation of the MPEC; in particular, the SCOC assumption introduced in Subsection 4.2.7 will play an important role in this approach. In both approaches, we establish that any accumulation point of the iterates produced by the algorithms must be a stationary point of MPEC provided that the point satisfies a strict complementarity assumption. The third class of algorithms is an extension of some locally convergent Newton methods for solving smooth nonlinear programs extended to a piecewise smooth setting; thus these algorithms are locally convergent in the sense that a closeness assumption is required on the initial iterate in order to guarantee convergence. Like their smooth counterpart, the piecewise smooth Newton algorithms have superlinear (and even quadratic) rates of convergence under mild assumptions. The chapter ends with a section that reports some preliminary computational results with a MATLAB implementation of PIPA; see Section 6.5.

The six chapters of the book are numbered from 1 to 6, the sections are denoted by decimal numbers of the type 2.3 (meaning Section 3 of Chapter 2). Many sections are further divided into subsections, some subsections are numbered, others are not. The numbered subsections are by decimal numbers following the section numbers; e.g., Subsection 1.3.1 means Chapter 1, Section 3, Subsection 1.

All definitions, results, and miscellaneous items are numbered consecutively within each section in the form 1.3.5, 1.3.6, meaning Items 5 and 6 in Section 3 of Chapter 1. All items are also identified by their types (e.g., 1.4.1 Proposition., 1.4.2 Remark.). When an item is referred to in the text, it is called out as Algorithm 5.2.1, Theorem 4.1.7, etc.

Equations are numbered consecutively in each section by (1), (2), etc. Any reference to an equation in the same section is by this number only, whereas equations in another section are identified by chapter, section, and equation. Thus (3.1.4) means Equation (4) in Section 1 of Chapter 3.

For reasons given in Section 1.1, we have seen that the general MPEC (1.1.1) is a very difficult constrained optimization problem to deal with. One reason is that the feasible region of the MPEC is defined implicitly as the solution set of a parametric variational inequality. To facilitate the design of solution procedures for MPEC, we need to represent its constraints in terms of a finite system of (nonlinear) equalities and inequalities, thus casting MPEC in the form of a standard nonlinear program. In Section 1.3, we have given various equivalent formulations of the constraints of the MPEC; in particular, the KKT forms (1.3.8) and (1.3.9) will play a major role in the study of the MPEC (1.1.1).

Once an MPEC has been formulated as a standard constrained optimization problem, such as (1.3.9), we may attempt to use any one of a number of nonlinear programming approaches to deal with it. Traditionally, penalty functions provide a powerful approach, both as a theoretical tool and as a computational vehicle, for the study of mathematical programs. Based on a recent exact penalty function theory of subanalytic optimization problems first obtained by Warga [283] and subsequently extended by Dedieu [57], this chapter develops a general exact penalization theory for the MPEC (1.1.1) under some mild continuity and subanalyticity assumptions on the objective and constraint functions of MPEC. We also obtain variations and improvements of the basic exact penalty function results. The principal tool that enables us to develop this theory is the theory of error bounds for systems of analytic inequalities, particularly those for quadratic inequalities.

This monograph deals with a class of constrained optimization problems which we call Mathematical Programs with Equilibrium Constraints, or simply, MPECs. Briefly, an MPEC is an optimization problem in which the essential constraints are defined by a parametric variational inequality or complementarity system. The terminology, MPEC, is believed to have been coined in [108]; the word “equilibrium” is adopted because the variational inequality constraints of the MPEC typically model certain equilibrium phenomena that arise from engineering and economic applications. The class of MPECs is an extension of the class of bilevel programs, also known as mathematical programs with optimization constraints, which was introduced in the operations research literature in the early 1970s by Bracken and McGill in a series of papers [34, 36, 37]. The MPEC is closely related to the economic problem of Stackelberg game [265] the origin of which predates the work of Bracken and McGill.

Our motivation for writing this monograph on MPEC stems from the practical significance of this class of mathematical programs and the lack of a solid basis for the treatment of these problems. Although there is a substantial amount of previous research on special cases of MPEC, no existing work provides such generality, depth, and rigor as the present study. Our intention in this monograph is to establish a sound foundation for MPEC that we hope will inspire further applications and research on this important problem.

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.

The consideration of variables sampling begun in the third chapter will be generalized in the present one to the treatment of several classes of reported data. Thus we continue with and extend a model of verification based upon statistical testing of quantitative measurement data, both those of inspector and inspectee, with the concomitant possibility of reaching the wrong conclusion: false alarm probabilities are finite. The main conclusion of the chapter, Theorem 4.2, is the variables sampling analog of Theorem 2.1.

The results to follow have been obtained primarily in connection with the analysis of inventory verification in nuclear fuel production and processing plants, where the material to be verified is extremely hazardous and valuable and is present in a wide variety of physical and chemical configurations. Prior to verification the material listed by the facility operator is stratified or classified according to characteristics such as accessibility, degree of purity, similarity of content, available measurement techniques, etc.

Stratification is an important prerequisite for the establishment of efficient sampling plans, not only for nuclear safeguards. For example Welsch (1992) mentions that, in the case of monitoring air pollution from power plants, seven different constituents have to be considered, each requiring its own measurement method.

Nevertheless, this chapter has been written without any one specific example in mind, the wide range of applications of the theory being, we feel, manifest.

The verification of compliance with a particular undertaking by use of on-site inspection is often influenced by considerations of timeliness. Brief but frequent visits may need to be made at regular or irregular intervals between more exhaustive inspections in order to obtain, with some degree of assurance, short notice indication of a possible violation. In Chapter 7 we shall be considering the problem of timely detection in the context of material accountancy under the Nuclear Weapons Non-Proliferation Treaty (NPT), where measurement uncertainties and the associated probabilities of false alarm play the decisive role. In the present chapter we treat a simpler NPT verification problem, very analogous to the attributes sampling problem of Section 2.2. In fact, this chapter might have been given the title Attributes Sampling over Time.

Interim inspections for timely detection generally involve excessive demands on travel expenses and manpower resources for the inspecting party, so it's of interest to look for inspection strategies which optimize timeliness under given constraints. Our optimization criterion will therefore be the expected time to detection of violation, rather than detection probability.

To illustrate, a fairy tale (Canty and Avenhaus (1991)):

The Inspector who got Something for Nothing

Once upon a time there was an inspector who wanted to spend more time with his family.

According to the Treaty on the Non-Proliferation of Nuclear Weapons (NPT), the safeguarding of fissionable material in the peaceful nuclear industry is carried out by the International Atomic Energy Agency as follows (IAEA (1972)): The operator of a safeguarded facility reports to the IAEA, through his national or multinational authority, the data required for the closing of a material balance. IAEA inspectors verify on-site the validity of the reported data by independent measurements, usually on a random sampling basis. The inspectorate then establishes with the aid of the reported data a material balance upon which it bases its technical conclusions.

In Chapter 7 we dealt with the decision-theoretical aspects of drawing conclusions from unfalsified material accountancy data, postponing the subjects of independent verification and deliberate falsification to the present chapter. Now, in the light of our experience with the subtleties of verification theory, we have first to face a rather fundamental question. Since the possibility of deliberate data falsification on the part of a facility operator intent on concealing a diversion of nuclear material cannot be excluded, and indeed ultimately justifies the very existence of a verification system, should his a a priori suspect data be included at all in the inspector's decision process?