Nonlinear Chance Constrained Problems: Optimality Conditions, Regularization and Solvers

Abstract

We deal with chance constrained problems with differentiable nonlinear random functions and discrete distribution. We allow nonconvex functions both in the constraints and in the objective. We reformulate the problem as a mixed-integer nonlinear program and relax the integer variables into continuous ones. We approach the relaxed problem as a mathematical problem with complementarity constraints and regularize it by enlarging the set of feasible solutions. For all considered problems, we derive necessary optimality conditions based on Fréchet objects corresponding to strong stationarity. We discuss relations between stationary points and minima. We propose two iterative algorithms for finding a stationary point of the original problem. The first is based on the relaxed reformulation, while the second one employs its regularized version. Under validity of a constraint qualification, we show that the stationary points of the regularized problem converge to a stationary point of the relaxed reformulation and under additional condition it is even a stationary point of the original problem. We conclude the paper by a numerical example.

Appendix

Together with the index sets defined in (6), we work with the following ones

$$\begin{aligned} I_{-+}(\bar{x},\bar{y}):= & {} {\left\{ i:g(\bar{x},\xi _i) < 0,\ 0<\bar{y}_i<1\right\} },\quad I_{-1}(\bar{x},\bar{y}) := {\left\{ i:g(\bar{x},\xi _i) < 0,\ \bar{y}_i=1\right\} },\\ I_{-0}(\bar{x},\bar{y}):= & {} {\left\{ i:g(\bar{x},\xi _i) < 0,\ \bar{y}_i=0\right\} },\quad I_{+0}(\bar{x},\bar{y}) := {\left\{ i:g(\bar{x},\xi _i) > 0,\ \bar{y}_i=0\right\} }. \end{aligned}$$

Lemma 6.1

Denote Z to be the feasible set of problem (5), fix any $(\bar{x}, \bar{y})\in Z$ with $\bar{y}\in Y(\bar{x})$. Let Assumption 2.1 be fulfilled at $\bar{x}$. Then we have $\hat{N}_Z(\bar{x},\bar{y})=N_1(\bar{x})\times N_2(\bar{y}),$ where

$$\begin{aligned} N_1(\bar{x})&= {\left\{ \sum _{i\in I_0(\bar{x}) }\lambda _i\nabla _x g(\bar{x},\xi _i) + \sum _{j\in J_0(\bar{x})} \mu _j\nabla h_j(\bar{x}): \begin{array}{ll} &{}\lambda _i\ge 0,\ i\in I_{01}(\bar{x},\bar{y})\cup I_{0+}(\bar{x},\bar{y})\\ &{}\lambda _i= 0,\ i\in I_{00}(\bar{x},\bar{y})\\ {} &{}\mu _j\ge 0,\ j\in J_0(\bar{x}) \\ \end{array} \right\} }, \\ N_2(\bar{y})&= {\left\{ \mu p+\gamma : \begin{array}{ll} &{}\mu \le 0\text { if }p^\top y = 1-\varepsilon ,\ \mu = 0\text { if }p^\top y > 1-\varepsilon \\ &{}\gamma _i\le 0,\ i\in I_{00}(\bar{x},\bar{y})\cup I_{-0}(\bar{x},\bar{y})\\ &{}\gamma _i=0,\ i\in I_{-+}(\bar{x},\bar{y})\cup I_{0+}(\bar{x},\bar{y})\\ &{}\gamma _i\ge 0,\ i\in I_{-1}(\bar{x},\bar{y})\cup I_{01}(\bar{x},\bar{y})\\ &{}\gamma _i\in {\mathbb R},\ i\in I_{+0}(\bar{x},\bar{y})\\ \end{array}\right\} }. \end{aligned}$$

Proof

For $I\subset I_{00}(\bar{x},\bar{y})$ define sets

$$\begin{aligned} Z_I^x:= & {} {\left\{ x:g(x,\xi _i)\le 0,\ i\in I\cup I_{01}(\bar{x},\bar{y}) \cup I_{0+}(\bar{x},\bar{y}),\ h_j(x)\le 0,\ j=1,\dots ,J \right\} } \\ Z_I^y:= & {} {\left\{ y: \begin{array}{ll} &{}p^\top y\ge 1-\varepsilon ,\ \ y_i=0,\ i\in I_{+0}(\bar{x},\bar{y})\cup (I_{00}(\bar{x},\bar{y}){\setminus } I),\\ &{}y_i\in [0,1],\ i\in I\cup I_{-1}(\bar{x},\bar{y})\cup I_{01}(\bar{x},\bar{y})\cup I_{-+}(\bar{x},\bar{y})\cup I_{0+}(\bar{x},\bar{y})\cup I_{-0}(\bar{x},\bar{y}) \end{array}\right\} } \end{aligned}$$

and observe that union of $Z_I:=Z_I^x\times Z_I^y$ with respect to all $I\subset I_{00}(\bar{x},\bar{y})$ locally around $(\bar{x},\bar{y})$ coincides with Z, which implies

$$\begin{aligned} \hat{N}_Z(\bar{x},\bar{y}) = \bigcap _{I\subset I_{00}(\bar{x},\bar{y})}\hat{N}_{Z_I}(\bar{x},\bar{y}) = \bigcap _{I\subset I_{00}(\bar{x},\bar{y})}\hat{N}_{Z_I^x}(\bar{x})\times \bigcap _{I\subset I_{00}(\bar{x},\bar{y})}\hat{N}_{Z_I^y}(\bar{y}). \end{aligned}$$

This means that we have

$$\begin{aligned} N_1(\bar{x}) = \bigcap _{I\subset I_{00}(\bar{x},\bar{y})}\hat{N}_{Z_I^x}(\bar{x}) = \hat{N}_{Z_\emptyset ^x}(\bar{x})\quad \text {and}\quad N_2(\bar{y}) = \bigcap _{I\subset I_{00}(\bar{x},\bar{y})}\hat{N}_{Z_I^y}(\bar{y}) = \hat{N}_{Z_{I_{00}(\bar{x}, \bar{y})}^y}(\bar{y}). \end{aligned}$$

The lemma statement them follows from [34, Proposition 3.4] and [27, Theorem 6.14]. $\square$