Alternating Minimization Algorithm for Polynomial Optimal Control Problems

Many optimal control problems can be cast into polynomial optimization problems through the discretization and conversion of expressions. The polynomial programming problem can further be transformed into a general quadratically constrained quadratic programing (QCQP) problem by introducing new variables and equality constraints. This paper develops an alternating minimization algorithm (AMA) to search for the optimal solution to a QCQP that is formulated as a rank-one constrained optimization problem. Based on the fact that a rank-one matrix is formed by two equivalent vectors, AMA alternatively solves each vector in sequence. Each sequential problem is a convex quadratic programming problem with linear constraints. A convergence analysis of AMA is provided. The efficacy of the proposed AMA is demonstrated by numerically solving two constrained optimal control problems where the conventional approach based on nonlinear programming experiences difficulty.