Automatic repair of convex optimization problems

Abstract

Given an infeasible, unbounded, or pathological convex optimization problem, a natural question to ask is: what is the smallest change we can make to the problem’s parameters such that the problem becomes solvable? In this paper, we address this question by posing it as an optimization problem involving the minimization of a convex regularization function of the parameters, subject to the constraint that the parameters result in a solvable problem. We propose a heuristic for approximately solving this problem that is based on the penalty method and leverages recently developed methods that can efficiently evaluate the derivative of the solution of a convex cone program with respect to its parameters. We illustrate our method by applying it to examples in optimal control and economics.

Appendix: Convex formulation

In the case that A is a constant while b and c are affine functions of $\theta$, we can write (9) as an equivalent convex optimization problem. In the linear case (i.e., when $\mathcal {K}= {\text{ R }}_+^n$), we can simply drop the strong duality requirement (which always holds in this case) and express (9) as

$$\begin{aligned} \begin{array}{ll} \text{ minimize } &{} r(\theta )\\ \text{ subject } \text{ to } ~ &{} Ax + s = b(\theta )\\ &{} A^Ty + c(\theta ) = 0\\ &{} s \in \mathcal {K}, \quad y \in \mathcal {K}^*. \end{array} \end{aligned}$$

For more general cones $\mathcal {K}$ (such as, e.g., the second order cone), a sufficient condition for strong duality is that there exist a feasible point in the interior of the cone. We can write this as, for example,

$$\begin{aligned} \begin{array}{ll} \text{ minimize } &{} r(\theta )\\ \text{ subject } \text{ to } ~ &{} Ax + s = b(\theta )\\ &{} A^Ty + c(\theta ) = 0\\ &{} s \in \mathop {{\mathbf{int }}}\limits \mathcal {K}, \quad y \in \mathcal {K}^{*}. \end{array} \end{aligned}$$

(14)

(We could similarly constrain $y\in \mathop {{\mathbf{int }}}\limits \mathcal {K}^{*}$ and $s\in \mathcal {K}$.)

In general, optimizing over open constraint sets is challenging and these problems may not even have an optimal point, but, in practice (and for well-enough behaved r, e.g., r continuous) we can approximate the true optimal value of (6) by approximating the open set $\mathop {{\mathbf{int }}}\limits \mathcal {K}$ as a sequence of closed sets $\mathcal {K}_\varepsilon \subseteq \mathop {{\mathbf{int }}}\limits \mathcal {K}$ such that $\mathcal {K}_\varepsilon \rightarrow \mathop {{\mathbf{int }}}\limits \mathcal {K}$ as $\varepsilon \downarrow 0$.