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First order constrained optimization algorithms with feasibility updates
TL;DR: First order algorithms for convex optimization problems where the feasible set is described by a large number of convex inequalities that is to be explored by subgradient projections are proposed.
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Abstract: We propose first order algorithms for convex optimization problems where the feasible set is described by a large number of convex inequalities that is to be explored by subgradient projections. The first algorithm is an adaptation of a subgradient algorithm, and has convergence rate $1/\sqrt{k}$. The second algorithm has convergence rate 1/k when (1) one has linear metric inequality in the feasible set, (2) the objective function is strongly convex, differentiable and has Lipschitz gradient, and (3) it is easy to optimize the objective function on the intersection of two halfspaces. This second algorithm generalizes Haugazeau's algorithm. The third algorithm adapts the second algorithm when condition (3) is dropped. We give examples to show that the second algorithm performs poorly when the objective function is not strongly convex, or when the linear metric inequality is absent.
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Citations
Complexity of linear minimization and projection on some sets
TL;DR: A motivation put forward in a large body of work on the Frank-Wolfe algorithm is the computational advantage of solving linear minimizations instead of projections, but the discussions supporting this advantage are often incomplete.
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Complexity of Linear Minimization and Projection on Some Sets
TL;DR: The Frank-Wolfe algorithm is a method for constrained optimization that relies on linear minimizations, as opposed to projections as discussed by the authors.However, the discussion supporting this advantage is often too succinct or incomplete.
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Set regularities and feasibility problems
TL;DR: In this article, the authors synthesize and unify notions of regularity, both of individual sets and of collections of sets, as they appear in the convergence theory of projection methods for consistent feasibility problems.
Set Regularities and Feasibility Problems
TL;DR: Several new characterizations of regularities are presented which shed light on the relations between seemingly different ideas and point to possible necessary conditions for local linear convergence of fundamental algorithms.
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John Darzentas
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Yurii Nesterov
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On Projection Algorithms for Solving Convex Feasibility Problems
TL;DR: A very broad and flexible framework is investigated which allows a systematic discussion of questions on behaviour in general Hilbert spaces and on the quality of convergence in convex feasibility problems.