Jérôme Bolte

First Order Methods Beyond Convexity and Lipschitz Gradient Continuity with Applications to Quadratic Inverse Problems

  • Jérôme Bolte
  • Shoham Sabach
  • Marc Teboulle
  • Yakov Vaisbourd
    • Categories Convex and Nonsmooth Optimization Tags , , , , , , , , ,

    We focus on nonconvex and nonsmooth minimization problems with a composite objective, where the differentiable part of the objective is freed from the usual and restrictive global Lipschitz gradient continuity assumption. This longstanding smoothness restriction is pervasive in first order methods (FOM), and was recently circumvent for convex composite optimization by Bauschke, Bolte and Teboulle, … Read more

    From error bounds to the complexity of first-order descent methods for convex functions

  • Jérôme Bolte
  • Trong Phong Nguyen
  • Juan Peypouquet
  • Bruce Suter
    • Categories Convex and Nonsmooth Optimization Tags , , , , , ,

    This paper shows that error bounds can be used as effective tools for deriving complexity results for first-order descent methods in convex minimization. In a first stage, this objective led us to revisit the interplay between error bounds and the Kurdyka-\L ojasiewicz (KL) inequality. One can show the equivalence between the two concepts for convex … Read more

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

  • Jérôme Bolte
  • Edouard Pauwels
    • Categories Applications - Science and Engineering, Nonlinear Optimization Tags , , , , , , , ,

    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

  • Hédy Attouch
  • Jérôme Bolte
  • Benar Fux Svaiter
    • Categories Nonlinear Optimization Tags , , , , , , , , , , , ,

    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

    Asymptotic expansions for interior penalty solutions of control constrained linear-quadratic problems

  • Felipe Álvarez
  • Jérôme Bolte
  • Joseph Frédéric Bonnans
  • Francisco J. Silva
    • Categories Other Topics

    We consider a quadratic optimal control problem governed by a nonautonomous affine differential equation subject to nonnegativity control constraints. For a general class of interior penalty functions, we show how to compute the principal term of the pointwise expansion of the state and the adjoint state. Our main argument relies on the following fact: If … Read more

    Generic identifiability and second-order sufficiency in tame convex optimization

  • Jérôme Bolte
  • Aris Daniilidis
  • Adrian Lewis
    • Categories Convex Optimization Tags , , ,

    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