tame optimization – Optimization Online

Majorization-minimization procedures and convergence of SQP methods for semi-algebraic and tame programs

Published: 2014/09/29

Applications - Science and Engineering, Nonlinear Optimization converging methods, extended sequential quadratic method, kkt points, kl inequality, moving balls method, nonlinear programming, sequential quadratic programming, sl1qp, tame optimization

In view of solving nonsmooth and nonconvex problems involving complex constraints (like standard NLP problems), we study general maximization-minimization procedures produced by families of strongly convex sub-problems. Using techniques from semi-algebraic geometry and variational analysis –in particular Lojasiewicz inequality– we establish the convergence of sequences generated by this type of schemes to critical points. The … Read more

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods

Published: 2010/12/22

Hédy Attouch

Jérôme Bolte

Benar Fux Svaiter

Nonlinear Optimization alternating direction method, block-coordinate methods, descent methods, forward-backward splitting, iterative thresholding, kurdyka-lojasiewicz inequality, nonconvex nonsmooth optimization, o-minimal structures, proximal algorithms, relative error, semi-algebraic optimization, sufficient decrease, tame optimization

In view of the minimization of a nonsmooth nonconvex function f, we prove an abstract convergence result for descent methods satisfying a sufficient-decrease assumption, and allowing a relative error tolerance. Our result guarantees the convergence of bounded sequences, under the assumption that the function f satisfies the Kurdyka-Lojasiewicz inequality. This assumption allows to cover a … Read more

Continuity of set-valued maps revisited in the light of tame geometry

Published: 2009/05/04

Aris Daniilidis

C.H. Jeffrey Pang

Nonsmooth Optimization aubin property, inner) continuity, outer, piecewise polyhedral, semialgebraic, set-valued map, tame optimization

Continuity of set-valued maps is hereby revisited: after recalling some basic concepts of variational analysis and a short description of the State-of-the-Art, we obtain as by-product two Sard type results concerning local minima of scalar and vector valued functions. Our main result though, is inscribed in the framework of tame geometry, stating that a closed-valued … Read more

Generic identifiability and second-order sufficiency in tame convex optimization

Published: 2009/01/20

Jérôme Bolte

Aris Daniilidis

Adrian Lewis

Convex Optimization o-minimal structure, partial smoothness, strong maximizer, tame optimization

We consider linear optimization over a fixed compact convex feasible region that is semi-algebraic (or, more generally, “tame”). Generically, we prove that the optimal solution is unique and lies on a unique manifold, around which the feasible region is “partly smooth”, ensuring finite identification of the manifold by many optimization algorithms. Furthermore, second-order optimality conditions … Read more