1,721,003 research outputs found
Projection free methods on product domains
Full text linkProjection-free block-coordinate methods avoid high computational cost per iteration, and at the same time exploit the particular problem structure of product domains. Frank–Wolfe-like approaches rank among the most popular ones of this type. However, as observed in the literature, there was a gap between the classical Frank–Wolfe theory and the block-coordinate case, with no guarantees of linear convergence rates even for strongly convex objectives in the latter. Moreover, most of previous research concentrated on convex objectives. This study now deals also with the non-convex case and reduces above-mentioned theory gap, in combining a new, fully developed convergence theory with novel active set identification results which ensure that inherent sparsity of solutions can be exploited in an efficient way. Preliminary numerical experiments seem to justify our approach and also show promising results for obtaining global solutions in the non-convex case
Frank–Wolfe and friends: a journey into projection-free first-order optimization methods
Full text linkInvented some 65 years ago in a seminal paper by Marguerite Straus-Frank and Philip Wolfe, the Frank–Wolfe method recently enjoys a remarkable revival, fuelled by the need of fast and reliable first-order optimization methods in Data Science and other relevant application areas. This review tries to explain the success of this approach by illustrating versatility and applicability in a wide range of contexts, combined with an account on recent progress in variants, improving on both the speed and efficiency of this surprisingly simple principle of first-order optimization
Ellipsoidal approach to box-constrained quadratic problems
No full textWe present a new heuristic for the global solution of box constrained quadratic problems, based on the classical results which hold for the minimization of quadratic problems with ellipsoidal constraints. The approach is tested on several problems randomly generated and on graph instances from the DIMACS challenge, medium size instances of the Maximum Clique Problem. The numerical results seem to suggest some effectiveness of the proposed approach
Mining for diamonds—Matrix generation algorithms for binary quadratically constrained quadratic problems
Full text linkIn this paper, we consider binary quadratically constrained quadratic problems and propose a new approach to generate stronger bounds than the ones obtained using the Semidefinite Programming relaxation. The new relaxation is based on the Boolean Quadric Polytope and is solved via a Dantzig–Wolfe Reformulation in matrix space. For block-decomposable problems, we extend the relaxation and analyze the theoretical properties of this novel approach. If overlapping size of blocks is at most two (i.e., when the sparsity graph of any pair of intersecting blocks contains either a cut node or an induced diamond graph), we establish equivalence to the one based on the Boolean Quadric Polytope. We prove that this equivalence does not hold if the sparsity graph is not chordal and we conjecture that equivalence holds for any block structure with a chordal sparsity graph. The tailored decomposition algorithm in the matrix space is used for efficiently bounding sparsely structured problems. Preliminary numerical results show that the proposed approach yields very good bounds in reasonable time
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