Abstract
Extended linear-quadratic programming arises as a flexible modeling scheme in dynamic and stochastic optimization, which allows for penalty terms and facilitates the use of duality. Computationally it raises new challenges as well as new possibilities in large-scale applications. Recent efforts have been focused on the fully quadratic case ([15] and [23]), while relying on the fundamental proximal point algorithm (PPA) as a shell of “outer” iterations when the problem is not fully quadratic. In this paper, we focus on the nonfully quadratic cases by proposing some new variants of the fundamental PPA. We first construct a continuously differentiable saddle function S(u, v) through infimal convolution in such a way that the optimal primal-dual pairs of the original problem are just the saddle points of S(u, v) on the whole space. Then the original extended linear-quadratic-programming problem reduces to solving the nonlinear equation ∇S(u, v)=0. We then embed the fundamental PPA and some of its previous variants in the framework of a Newton-like iteration for this equation. After revealing the local quadratic structure of S near the solution, we derive new extensions of the fundamental PPA. In numerical tests, the modified iteration scheme based on the quasi-Newton update formula outperforms the fundamental PPA considerably.
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Zhu, C. Modified proximal point algorithm for extended linear-quadratic programming. Comput Optim Applic 1, 185–205 (1992). https://doi.org/10.1007/BF00253806
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DOI: https://doi.org/10.1007/BF00253806