Theoretical and numerical challenges introduced into the NLP by outer convexification of constraints depending on binary or integer controls have been exemplarily studied in this section. Our findings clearly justify the need for deeper investigation of the combinatorial structure of such NLPs in order to take the structure into account in the design of a tailored SQP algorithm.
The findings of the first section lead us to investigate NLPs with constraints treated by outer convexification from a different point of view. In this section we introduce the problem class of Mathematical Programs with Vanishing Constraints (MPVCs), highly challenging nonconvex problems that have only recently attracted increased research interest. This problem class provides a framework for analysis of the shortcomings of the standard Lagrangian approach at solving MPVCs, and yields new stationarity conditions for a separated, i.e., non–multiplicative formulation of the vanishing constraints that will in general be much more well conditioned. Pioneering work on the topic of MPVCs was brought forward only recently by [3, 106, 107]. In these works, MPVCs were shown to constitute a challenging class of problems, as standard constraint qualifications such as LICQ and MFCQ turn out to be violated for most problem instances. In the thesis [105] an exhaustive treatment of the theoretical properties of MPVCs can be found. The theoretical results presented in this section are a summary of those parts of
[109] and the aforementioned works that are required for the numerical treatment of NLPs with constraints treated by outer convexification.
5.3.1 Problem Formulations
Definition 5.3 (Nonlinear Program with Vanishing Constraints)
The following NLP with l¾ 1 complementary inequality constraints min
x ∈Rn f (x) (5.9)
s. t. gi(x ) · hi(x ) ¾ 0, 1¶ i ¶ l, h(x) ¾ 0
is called a Mathematical Program with Vanishing Constraints (MPVC). 4 The objective function f :Rn → R and the constraint functions g
i, hi :R → R, 1 ¶ i ¶ l are expected to be at least twice continuously differentiable. The domain of feasibility of problem (5.9) may be further restricted by a standard nonlinear constraint function c(x) ½ 0, including equality constraints. Such constraints do not affect the theory and results to be presented in this chapter, and we omit them for clarity of exposition.
Constraint Logic Reformulation
It is easily observed that the constraint functions gi(x ) do not impact the question of feasibility of an arbitrary point x ∈ Rniff h
i(x ) = 0 holds. The constraint gi is the said to have vanished in the point x . This observation gives rise to the following constraint logic reformulation.
Remark 5.6 (Constraint Logic Reformulation for MPVCs)
Problem (5.9) can be equivalently cast as an NLP with m logic constraints,
min
x ∈Rn f (x) (5.10)
s. t. hi(x ) > 0 =⇒ gi(x ) ¾ 0, 1¶ i ¶ m, h(x) ¾ 0.
The logical implication though appears to be unsuitable for derivative based numerical meth- ods. We will however see in the sequel of this chapter that it can be realized in the framework of an active set method for Quadratic Programming.
Equilibrium Constraint Reformulation
MPVCs can be related to the well studied class of Mathematical Programs with Equilibrium Constraints (MPECs) through the following reformulation.
Remark 5.7 (MPEC Reformulation for MPVCs)
variables, min x ∈Rn f (x) (MPEC) s. t. ξi· hi(x ) = 0, 1¶ i ¶ l, h(x) ¾ 0, ξ¶ 0, g (x) − ξ ¾ 0.
Although this opens up the possibility of treating (5.9) using preexisting MPECs algorithms designed for interior point methods by e.g. [17, 138, 172] and for active set methods by e.g. [75, 111], this reformulation suffers from the fact that the additionally introduced vector ξ of slack variables is undefined if hi(x?) = 0 holds for some index i in an optimal solution x?. In fact, MPECs are known to constitute an even more difficult class than MPVCs, as both LICQ and MFCQ are violated in every feasible point [47]. Consequentially, standard NLP sufficient conditions for optimality of x?as presented in section 3.1.3 do not hold. If on the other hand hi(x?) 6= 0 for all indices 1 ¶ i ¶ l, we could as well have found the same solution x? by including all vanishing constraints gi as standard constraints, which allows to treat problem (5.9) as a plain NLP.
Relaxation Method
It has frequently been proposed to solve problem (5.9) by embedding it into a family of perturbed problems. The vanishing constraint that causes violation of constraint qualifications is relaxed,
min
x ∈Rn f (x) (MPVC(τ))
s. t. gi(x ) · hi(x ) ¾ −τ, 1¶ i ¶ l, h(x) ¾ 0
where the scalar τ¾ 0 is the relaxation parameter. For τ > 0, problem (MPVC(τ)) is regular in its solution. The level of exactness of this relaxation is investigated e.g. in [109].
Embedding Relaxation Method
A more general relaxation of the vanishing constraint gi(x ) · hi(x ) ¾ 0 is proposed in [105, 109] that works by embedding it into a function ϕ satisfying
ϕ(a, b) = 0 ⇐⇒ b ¾ 0, a · b ¾ 0,
and introducing a relaxation parameter τ together with a family of functions ϕτ(a, b) for which it holds that ϕτ(a, b) → ϕ(a, b) for τ → 0. It is argued that smooth functions ϕ necessarily lead to lack of LICQ and MFCQ, and the use of a family of nonsmooth functions is proposed and convergence results are obtained similar in spirit to those of [174, 191] for MPECs.
h g
F
(a) Feasible set of original problem (5.9).
h
−ξ
F
(b) Feasible set of MPEC reformulation.
F
τ→ 0+
h g
(c) Feasible set of relaxed problem (MPVC(τ)), τ = 0.1.
Figure 5.5: Feasible sets of various reformulations of problem (5.9).
Remark 5.8 (Ill–Conditioning of Reformulations)
The ill–conditioning investigated in section 5.2.3, inherent to the multiplicative formulation of the vanishing constraint, is not resolved by either of the two relaxation methods.
Remark 5.9 (Relaxation of Constraints)
Relaxation methods require the problem’s objective and constraint functions to be valid out- side the feasible set originally described by problem (5.9). As we will see in chapter 9, this may lead to evaluations of model functions outside the physically meaningful domain. If re- laxation methods are employed, it is important to allow for such evaluations in a way that does not prevent the true solution from being found.