We deal with the conditions which ensure exact penalization in stochastic programming problems under finite discrete distributions. We give several sufficient conditions for problem calmness including graph calmness, existence of an error bound, and generalized Mangasarian-Fromowitz constraint qualification. We propose a new version of the theorem on asymptotic equivalence of local minimizers of chance constrained problems and problems with exact penalty objective. We apply the theory to a problem with a stochastic vanishing constraint.

Constraints of real life optimization problems often depend on a random vector. If we know its probability distribution or if we have a good estimate, we can use various stochastic programming formulations to solve the problem and to get a highly reliable solution with respect to the realizations of the random vector. In chance constrained programming problems, a probability for fulfilling the random constraints is prescribed; see Prékopa [

In this paper, we investigate an approach which uses penalized random constraints incorporated into the objective function. It can be verified that under some conditions we obtain an optimal solution with desirable reliability by increasing the penalty parameter and solving the resulting problems. It can be even shown that under mild conditions the penalty approach and the chance constrained problem are asymptotically equivalent; see Branda [

There is an extensive development of the exact penalty methods for deterministic optimization in recent years. Antczak [

The paper is organized as follows. The basic notation and problem formulations are proposed in Section

Let

An important role is put to the problem where all random constraints are fulfilled. To simplify the notation, we set

We will employ two concepts of calmness. The first is related directly to the perturbed problem (

Let

Note that if the problem (

The following proposition was formulated by Burke [

Let

The form of the penalty problem follows from the theorem:

In this section, we give necessary conditions to ensure calmness of the problem (

Let

local Lipschitz property at

Aubin property at

local upper Lipschitz property at

(graph) calmness at

It can be seen directly from the definitions that the calmness is the weakest of the conditions and is implied by the local upper Lipschitz property or by the Aubin property, whereas these two conditions are ensured by the local Lipschitz property.

For

Let

See Hoheisel et al. [

The preposition shows that sufficient conditions for calmness of the constrained set are desirable. Global graph calmness is ensured in the following important case.

The set-valued mapping

See Rockafellar and Wets [

In general, it is not possible to obtain such strong result. Thus, we have to focus on conditions which ensure the calmness at least locally. These conditions are based on constraint qualification; see, for example, Bazaraa et al. [

Let

The relevance of the GMFCQ to our analysis is confirmed by the following proposition.

Let

For the first part see Flegel et al. [

Another important property, which is equivalent to graph calmness, is the existence of a local error bound of the constraint set.

Calmness of the constraints system (

See Hoheisel et al. [

In this section, we propose a new version of the theorem which concerns asymptotic equivalence of chance constrained problems and problems with penalty objective. We use the notion of optimal values for local optimal values; that is, the vector of decision variables

One considers the chance constrained problem (

the problem (

the GMFCQ is satisfied at the local solution

Then for any prescribed

Moreover, bounds on the local optimal value

The proof is analogous to Theorem 3 proposed by Branda [

Let

Let

In this paper, we have discussed conditions which ensure exact penalization in stochastic programming. We have started with a general calmness condition of the underlying problem, which is equivalent to the exact penalty property. However, this condition cannot be verified directly; therefore necessary conditions have been proposed. We have shown that the local Lipschitz continuity of the objective function together with the calmness of the constraint set, which is implied, for example, by generalized Mangasarian-Fromovitz constraint qualification, is sufficient. These findings have enabled us to propose a new version of the theorem concerning asymptotic equivalence of the chance constrained problems and the problems with exact penalty objective, which is the main contribution of this paper. The theoretical results have been applied to the problem with a stochastic vanishing constraint where a sufficient condition for fulfilling the GMFCQ has been formulated.

The author declares that there is no conflict of interests regarding the publication of this paper.

This work was supported by the Czech Science Foundation under the Grant GP13-03749P.