# Convex minimization under Lipschitz constraints

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## Abstract

We consider the problem of minimizing a convex function*f(x)* under Lipschitz constraints*f*_{ i }(*x*)≤0,*i*=1,...,*m*. By transforming a system of Lipschitz constraints*f*_{ i }(*x*)≤0,*i*=*l*,...,*m*, into a single constraints of the form*h(x)*-∥*x*∥^{2}≤0, with*h*(·) being a closed convex function, we convert the problem into a convex program with an additional reverse convex constraint. Under a regularity assumption, we apply Tuy's method for convex programs with an additional reverse convex constraint to solve the converted problem. By this way, we construct an algorithm which reduces the problem to a sequence of subproblems of minimizing a concave, quadratic, separable function over a polytope. Finally, we show how the algorithm can be used for the decomposition of Lipschitz optimization problems involving relatively few nonconvex variables.

## Key Words

Global optimization convex minimization Lipschitz constraints reverse convex constraints## Preview

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