Edouard Pauwels

Majorization-Minimization Procedures and Convergence of SQP Methods for Semi-Algebraic and Tame Programs

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 subproblems. Using techniques from semi-algebraic geometry and variational analysis—in particular Łojasiewicz inequality—we establish the convergence of sequences generated by these types of schemes to critical points. The broad applicability of this process is illustrated in the context of NLP. In that case, critical points coincide with KKT points. When the data are semi-algebraic or real analytic our method applies (for instance) to the study of various sequential quadratic programming (SQP) schemes: the moving balls method, the penalized SQP method and the extended SQP method. Under standard qualification conditions, this provides—to the best of our knowledge—the first general convergence results for general nonlinear programming problems. We emphasize the fact that, unlike most works on this subject, no second-order conditions and/or convexity assumptions whatsoever are made. Rate of convergence are shown to be of the same form as those commonly encountered with first-order methods.

The Cyclic Block Conditional Gradient Method for Convex Optimization Problems

In this paper we study the convex problem of optimizing the sum of a smooth function and a compactly supported nonsmooth term with a specific separable form. We analyze the block version of the generalized conditional gradient method when the blocks are chosen in a cyclic order. A global sublinear rate of convergence is established for two different stepsize strategies commonly used in this class of methods. Numerical comparisons of the proposed method to both the classical conditional gradient algorithm and its random block version demonstrate the effectiveness of the cyclic block update rule.

Linear conic optimization for inverse optimal control

We address the inverse problem of Lagrangian identification based on trajecto-ries in the context of nonlinear optimal control. We propose a general formulation of the inverse problem based on occupation measures and complementarity in linear programming. The use of occupation measures in this context offers several advan-tages from the theoretical, numerical and statistical points of view. We propose an approximation procedure for which strong theoretical guarantees are available. Finally, the relevance of the method is illustrated on academic examples.

On Fienup Methods for Sparse Phase Retrieval

Alternating minimization, or Fienup methods, have a long history in phase retrieval. We provide new insights related to the empirical and theoretical analysis of these algorithms when used with Fourier measurements and combined with convex priors. In particular, we show that Fienup methods can be viewed as performing alternating minimization on a regularized nonconvex least-squares problem with respect to amplitude measurements. Furthermore, we prove that under mild additional structural assumptions on the prior (semialgebraicity), the sequence of signal estimates has a smooth convergent behavior toward a critical point of the nonconvex regularized least-squares objective. Finally, we propose an extension to Fienup techniques, based on a projected gradient descent interpretation and acceleration using inertial terms. We demonstrate experimentally that this modification combined with an

Primal and dual predicted decrease approximation methods

\ell _1

Positivity certificates in optimal control

prior constitutes a competitive approach for sparse phase retrieval.

Towards Ultrafast Subwavelength Microscopy

We introduce the notion of predicted decrease approximation (PDA) for constrained convex optimization, a flexible framework which includes as special cases known algorithms such as generalized conditional gradient, proximal gradient, greedy coordinate descent for separable constraints and working set methods for linear equality constraints with bounds. The new scheme allows the development of a unified convergence analysis for these methods. We further consider a partially strongly convex nonsmooth model and show that dual application of PDA-based methods yields new sublinear convergence rate estimates in terms of both primal and dual objectives. As an example of an application, we provide an explicit working set selection rule for SMO-type methods for training the support vector machine with an improved primal convergence analysis.

Single-shot ptychography & sparsity-based subwavelength ptychography

We propose a tutorial on relaxations and weak formulations of optimal control with their semidefinite approximations. We present this approach solely through the prism of positivity certificates which we consider to be the most accessible for a broad audience, in particular in the engineering and robotics communities. This simple concept allows to express very concisely powerful approximation certificates in control. The relevance of this technique is illustrated on three applications: region of attraction approximation, direct optimal control and inverse optimal control, for which it constitutes a common denominator. In a first step, we highlight the core mechanisms underpinning the application of positivity in control and how they appear in the different control applications. This relies on simple mathematical concepts and gives a unified treatment of the applications considered. This presentation is based on the combination and simplification of published materials. In a second step, we describe briefly relations with broader literature, in particular, occupation measures and Hamilton-Jacobi-Bellman equation which are important elements of the global picture. We describe the Sum-Of-Squares (SOS) semidefinite hierarchy in the semialgebraic case and briefly mention its convergence properties. Numerical experiments on a classical example in robotics, namely the nonholonomic vehicle, illustrate the concepts presented in the text for the three applications considered.

Linear conic optimization for nonlinear optimal control

We overcome two critical limitations in ptychography, a currently scanning-based diffraction-limited imaging technique. We demonstrate sub-wavelength ptychography using two approaches (subwavelength precision of the probe beam positions and sparsity prior) and also demonstrate single-shot ptychography.

On Fienup Methods for Regularized Phase Retrieval.

We overcome two major limitations in ptychography - a powerful scanning coherent diffraction imaging technique. We demonstrate single-shot ptychography, overcoming the scanning temporal resolution limit and demonstrate sparsity-based subwavelength ptychography, overcoming the Abbe resolution limit.