Exact Penalty Method for Variationally Coherent Stochastic Programming Problems

Abstract

The paper concerns optimization problems with general equality and inequality constraints and with constraints expressed by a convex set. In order to solve these problems, the general constraints are treated by an exact penalty functions while the others by mirror descent approach. The paper introduces a constraint qualification condition under which the solution of the optimization problem with an exact penalty function and constraints defined by the convex set is a solution of the original problem with constraints. The paper extends results on exact penalty functions to the case when together with general equality and inequality constraints additional constraints defined by a convex set are present. In order to solve the optimization problems with exact penalty functions, a mirror descent algorithm is proposed. It is assumed that instead of using gradients of functions defining constrained optimization problems, their stochastic approximations can be applied. The paper establishes global convergence of the proposed method under the assumption that applied exact penalty functions lead to variationally coherent optimization problems. Since exact penalty functions are not differentiable, the concept of variationally coherent problems is extended to the problems defined by functions exhibiting Clarke's generalized gradients. The behavior of the proposed method is illustrated by some numerical examples.

Figures (5)

• Figure 1: 3D visualization of the well-behaved $l_2$ penalty function for constraints $x_1 \leq 0,\ x_2-x_1 \leq 0$. The distinctive feature of this formulation is that the function is locally differentiable outside the feasible set. At the same time it models well a sharp minimum defined by constraints.
• Figure 2: Example trajectories generated by the algorithm. The white $+$ sign denotes the starting point of the algorithm, and $\times$ denotes the ending point. The red rectangle is the boundary of the set $X$, onto which the algorithm projects every iterate. The green line denotes all points, for which the penalty function is equal to zero. The orange line shows the trajectory taken by the algorithm.
• Figure 3: Convergence example for Rosenbrock's function $f(x) = \sum_{i=1}^{n-1} [100(x_{i+1} - x_i^2)^2 + (1 - x_i)^2]$, for $x \in {\mathbb{R}}^n$ and $n \in \{4, 8, 16, 32\}$. Both axes are scaled logarithmically. The mirror function is an identity function. The stochastic gradient oracle takes gradient of one random index at each step.
• Figure 4: Evolution of the 1D adaptive penalty method. Top: The penalty function $P_{p_k}(x)$ reshaping as $p_k$ increases (purple to yellow). Middle: Subgradient norm over time, oscillating at the non-smooth minimum. Bottom: Convergence of the total penalty function value.
• Figure 5: Comparison of the adaptive algorithm using a smooth penalty outside the feasible set ($\beta$-norm, left) versus a non-smooth penalty ($l_1$, right). The smooth formulation allows the gradient to vanish, successfully triggering the penalty update.

Theorems & Definitions (28)

• Definition 1: Constraint Qualification -- (CQ)
• Lemma 2
• proof
• Theorem 3
• Theorem 4
• proof
• Theorem 5
• proof
• Definition 6: Generalized Directional Derivative
• Definition 7: Clarke's Generalized Gradient