Constraint Qualification And Optimality Conditions For Mathematical Programs With Equilibrium Constraints

In recent years, mathematical programs with equilibrium constaints has been widely used in economic equilibrium, engineering, transport network de-sign and many other areas. As most usual constraint qualifications for standard nonlinear programming are no longer satisfied in this kind of problems, the well-known Karush-Kuhn-Tucker conditon may not be the first-order neces-sary optimality condition. These make the problem more complex. Therefore, it has been hot issues during the last few years to find new tailored constraint qualifications under which a local minimizer of this program satisfies some first-order optimality conditions.This is a survey of constraint qualifications and optimality conditions for mathematical programs with equilibrium constraints. We mainly introduce the new tailored constraint qualifications and corresponding optimality conditions introduced by M.L.Flegel and C.Kanzow since the year of 2000. A discussion of the relationship of these new constraint qualifications would be made. It also involves the concept of stationary points which can be made use of to ex-press optimality conditions. Because the Fritz John condition doesn't need any constraint qualifications, we also introduce a new result about the enhenced Fritz John condition proved by C.Kanzow and A.Vath.We consider the following problem: This kind of problem has its origin in finding the equilibrium solution of Stack-elberg games. We call it mathematical programs with equilibrium constraints, MPEC for short. 1.§3.1 We introduce some constraint qualifications for MPEC and the re-lationship between them. The MPEC tailored constraint qualification MPEC-ACQ will be introduced in§3.1.1. The GCQ will be introduced in§3.1.2. In§3.1.3. it involves other constraint qualifications and the relationship between them.1.1. M.L.Flegel and C.Kanzow defined a new cone named TMPEClin(z*) and gave the relationship between TMPEClin(z*) and T(z*).Theorem 1 For any feasible point z* of MPEC, we have They defined a new suitable constraint qualification called MPEC-ACQ: and they provided a sufficient condition for MPEC-ACQ in [9].Theorem 2 Let z* be a feasible point of MPEC. If g, h, G, H are all linear funtions, then MPEC-ACQ holds.1.2. In [3], M.L.Flegel and C.Kanzow pointed that although most con-strait qualification for standard nonlinear programming are violated in MPEC, the GCQ has a good chance to hold. They gave the following result.Theorem 3 Let z* be a feasible point of MPEC. If Al assumption holds, then conv(T(z*)) is closed. i.e. we have They also proved the equivalent form of GCQ in MPEC.1.3. M.L.Flegel provided other constraint qualifications and discussed the relationship between them in details in [12]. Theorem 4 If a feasible point z* of MPEC satisfies MPEC-LICQ, it satisfies MPEC-MFCQ.Theorem 5 If a feasible point z* of MPEC satisfies MPEC-MFCQ, it satisfies MPEC-ACQ.Theorem 6 If a feasible point z* of MPEC satisfies MPEC-LICQ, it satisfies GCQ.Theorem 7 If a feasible point z* of MPEC satisfies both MPEC-ACQ and A2 assumption,then it satisfies GCQ.Therefore, we have MPEC-LICQ(?)MPEC-MFCQ(?)MPEC-ACQ, MPEC-LICQ(?)GCQ, MPEC-ACQ+A2(?)GCQ.2.§3.2 We introduce optimality conditions for MPEC under suitable constraint qualifications. As the KKT point may no longer be the optimality condition for MPEC, stationary points which can be used to express optimality condition for MPEC are need. At last, we introduce a new result about the enhenced Fritz John condition proved by C.Kanzow and A.Vath.2.1. M.L.Flegel and C.Kanzow gave the optimality condition for MPEC under GCQ in [12].Theorem 8 Let z* be the local minimum point of MPEC, if GCQ is satisfied at z*, then z* is strong stationary.J.J.Ye proved in [4] that under MPEC-ACQ, a local minimum point of MPEC must be M-stationary. Theorem 9 Let z* be the local minimum point of MPEC.if MPEC-ACQ is satisfied at z*,then z* is M-stationary.2.2 C.Kanzow and A.Vath proved the enhenced Fritz John condition for MPEC in[5].Theorem 10 Let 2.be the local minimum point of MPEC.then there are multipliersλ*=(λf,λg,λh,λG,λH),s.t.(a)(b)(c)λf,λg,λh,λG,λH are not all equal to zero.(d) Ifλg,λh,λG,λH are not all equal to zero,then there is a sequence (zk)→z*, s.t.for all k∈N f(zk)0,thenλiggi(zk)>0, ifλjh≠0,thenλihhi(zk)>0, ifλiG≠0,thenλiGGi(zk)<0, ifλiH≠0,thenλiHHi(zk)<0,...