Multiple Cuts in Separating Plane Algorithms

This paper presents an extended version of the separation plane algorithms for subgradientbased finite-dimensional nondifferentiable convex blackbox optimization. The extension introduces additional cuts for epigraph of the conjugate of objective function which improve the convergence of the algorithm. The case of affine cuts is considered in more details and it is shown that it requires solution of an auxiliary convex subproblem the dimensionality of which depends on the number of additional cuts and can be kept arbitrary low. Therefore algorithm can make use of the efficient algorithms of low-dimensional nondifferentiable convex optimization which overcomes known computational complexity bounds for the general case. keywords: convex optimization, conjugate function, cutting plane, separating plane, center of gravity algorithm Introduction and Notations We consider a finite-dimensional nondifferentiable convex optimization (NCO) problem min x∈E f(x) = f⋆ = f(x ), x ∈ X⋆ , (1) where E denotes a finite-dimensional space of primal variables and f : E → R is a finite convex function, not necessarily differentiable. As we are interested in computational issues related to solving (1) mainly we assume that this problem is solvable and has nonempty set of solutions X⋆. This problem enjoys a considerable popularity due to its important theoretical properties and numerous applications in large-scale structured optimization, Lagrange relaxation in discrete optimization, exact penalization in constrained optimization, and others. This led to the development of different algorithmic ideas, starting with the subgradient algorithm due to Shor [1] and Polyak [2] and followed by cutting plane [3], conjugate subgradient [4], bundle methods [13], ellipsoid and space dilatation [5, 6, 7], ǫ-subgradient methods [8, 9], V U -methods [10] and ∗This work was partially supported by RFBR grant 13-07-1210