A Computation Approach to Chance Constrained Optimization of Boundary-Value Parabolic Partial Differential Equation Systems

This work studies chance constrained optimization of boundary-value parabolic partial differential equations (CCPDE) with random data, where the PDE model is treated as equality constraint and chance constraints are imposed on inequality constraints involving state variables. Since such a CCPDE problem is generally non-smooth, non-convex and difficult to solve directly, we use our recently proposed smoothing approximation method to solve the problem. As a result, the probability function of the chance constraints is approximated in two different ways by a family of differentiable functions. This leads to two smooth parametric optimization problems IAτ and OAτ, where the feasible sets of IAτ are always subsets (inner approximation) and the feasible sets of OAτ always supersets (outer approximation). The feasible sets of IAτ (resp. OAτ ) converge asymptotically to the feasible set of the CCPDE. Moreover, any limit point of a sequence of optimal solutions of IAτ (resp. OAτ ) is a stationary point of CCPDE. The viability of the approximation approach is numerically demonstrated by optimal thermal cancer treatment as a case study.