Abstract
In the present paper we propose to rewrite a nonsmooth problem subjected to convex constraints as an unconstrained problem. We show that this novel formulation shares the same global and local minima with the original constrained problem. Moreover, the reformulation can be solved with standard nonsmooth optimization methods if we are able to make projections onto the feasible sets. Numerical evidence shows that the proposed formulation compares favorably against state-of-art approaches. Code can be found at https://github.com/jth3galv/dfppm.
Highlights
In this paper, we consider the optimization of a nonsmooth function f ∶ Rn → R over a closed convex set, namely min f (x) s.t. x ∈ X. (1)We assume that f is locally Lipschitz continuous and that first order information is unavailable or impractical to obtain.The aim of the optimization, for nonsmooth problems, is to find Clarke-stationary points [6, 9].1 3 Vol.:(0123456789)
In this work we propose a novel way of treating convex constraints that is not based on penalty functions
Since f and ftake the same values on X it holds that any global minimum of the modified problem which belongs to the feasible set is o global minimum of the original problem
Summary
We consider the optimization of a nonsmooth function f ∶ Rn → R over a closed convex set, namely min f (x) s.t. x ∈ X. Several approaches, based on the pattern search methods dating back to [13], have been developed for bound and linearly constrained problems in [16] and [17] and more general type of constraints in [18]. A second main approach, proposed in [10], is instead based on an exact penalty function In this case, the feasible set is expressed by a possibly nonsmooth set of inequalities g ∶ Rn → Rm and the original problem is replaced by the penalized version, for a given ε > 0, min x f (x).
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