Primal-Dual Projected Gradient Algorithms for Extended Linear-Quadratic Programming

Many large-scale problems in dynamic and stochastic optimization can be modeled with extended linear-quadratic programming, which admits penalty terms and treats them through duality. In general, the objective functions in such problems are only piecewise smooth and must be minimized or maximized relative to polyhedral sets of high dimensionality. This paper proposes a new class of numerical methods for "fully quadratic" problems within this framework, which exhibit second-order nonsmoothness. These methods, combining the idea of finite-envelope representation with that of modified gradient projection, work with local structure in the primal and dual problems simultaneously, feeding information back and forth to trigger advantageous restarts. Versions resembling steepest descent methods and conjugate gradient methods are presented. When a positive threshold of $\varepsilon $-optimality is specified, both methods converge in a finite number of iterations. With threshold 0, it is shown under mild assumptions that the steepest descent version converges linearly, while the conjugate gradient version still has a finite termination property. The algorithms are designed to exploit features of primal and dual decomposability of the Lagrangian, which are typically available in a large-scale setting, and they are open to considerable parallelization.