In this paper we develop a subgradient optimization methodology for convex multiparametric nonlinear programs. We define a parametric subgradient, extend some classical optimization results to the multiparametric case, and design a subgradient algorithm that is shown to converge under traditional conditions. We use this algorithm to solve two illustrative example problems and demonstrate its ac...

The subgradient projector is of considerable importance in convex optimization because it plays the key role in Polyak’s seminal work — and the many papers it spawned — on subgradient projection algorithms for solving convex feasibility problems. In this paper, we offer a systematic study of the subgradient projector. Fundamental properties such as continuity, nonexpansiveness, and monotonicity...

We introduce a generalised subgradient for law-invariant closed convex risk measures on L and establish its relationship with optimal risk allocations and equilibria. Our main result gives sufficient conditions ensuring a non-empty generalised subgradient.

The subgradient method is used frequently to optimize dual functions in Lagrangian relaxation for separable integer programming problems. In the method, all subproblems must be solved optimally to obtain a subgradient direction. In this paper, the surrogate subgradient method is developed, where a proper direction can be obtained without solving optimally all the subproblems. In fact, only an a...

We introduce a generalised subgradient for law-invariant closed convex risk measures on L and establish its relationship with optimal risk allocations and equilibria. Our main result gives sufficient conditions ensuring a non-empty generalised subgradient.

A study of the convergence properties of spectral projected subgradient method is presented and the convergence is shown. The convergence is based on spectral projected gradient approach. Some updates of the spectral projected subgradient are described.