Linear conic optimization for nonlinear optimal control

Infinite-dimensional linear conic formulations are described for nonlinear optimal control problems. The primal linear problem consists of finding occupation measures supported on optimal relaxed controlled trajectories, whereas the dual linear problem consists of finding the largest lower bound on the value function of the optimal control problem. Various approximation results relating the original optimal control problem and its linear conic formulations are developed. As illustrated by a couple of simple examples, these results are relevant in the context of finite-dimensional semidefinite programming relaxations used to approximate numerically the solutions of the infinite-dimensional linear conic problems.Comment: Submitted for possible inclusion as a contributed chapter in S. Ahmed, M. Anjos, T. Terlaky (Editors). Advances and Trends in Optimization with Engineering Applications. MOS-SIAM series, SIAM, Philadelphi

Generic Fr{\'e}chet stationarity in constrained optimization

Minimizing a smooth function f on a closed subset C leads to different notions of stationarity: Fr{\'e}chet stationarity, which carries a strong variational meaning, and criticallity, which is defined through a closure process. The latter is an optimality condition which may loose the variational meaning of Fr{\'e}chet stationarity in some settings. We show that, while criticality is the appropriate notion in full generality, Fr{\'e}chet stationarity is typical in practical scenarios. This is illustrated with two main results, first we show that if C is semi-algebraic, then for a generic smooth semi-algebraic function f , all critical points of f on C are actually Fr{\'e}chet stationary. Second we prove that for small step-sizes, all the accumulation points of the projected gradient algorithm are Fr{\'e}chet stationary, with an explicit global quadratic estimate of the remainder, avoiding potential critical points which are not Fr{\'e}chet stationary, and some bad local minima

Sorting out typicality with the inverse moment matrix SOS polynomial

International audienceWe study a surprising phenomenon related to the representation of a cloud of data points using polynomials. We start with the previously unnoticed empirical observation that, given a collection (a cloud) of data points, the sublevel sets of a certain distinguished polynomial capture the shape of the cloud very accurately. This distinguished polynomial is a sum-of-squares (SOS) derived in a simple manner from the inverse of the empirical moment matrix. In fact, this SOS polynomial is directly related to orthogonal polynomials and the Christoffel function. This allows to generalize and interpret extremality properties of orthogonal polynomials and to provide a mathematical rationale for the observed phenomenon. Among diverse potential applications, we illustrate the relevance of our results on a network intrusion detection task for which we obtain performances similar to existing dedicated methods reported in the literature

Predicting drug side-effect profiles: a chemical fragment-based approach

<p>Abstract</p> <p>Background</p> <p>Drug side-effects, or adverse drug reactions, have become a major public health concern. It is one of the main causes of failure in the process of drug development, and of drug withdrawal once they have reached the market. Therefore, <it>in silico </it>prediction of potential side-effects early in the drug discovery process, before reaching the clinical stages, is of great interest to improve this long and expensive process and to provide new efficient and safe therapies for patients.</p> <p>Results</p> <p>In the present work, we propose a new method to predict potential side-effects of drug candidate molecules based on their chemical structures, applicable on large molecular databanks. A unique feature of the proposed method is its ability to extract correlated sets of chemical substructures (or chemical fragments) and side-effects. This is made possible using sparse canonical correlation analysis (SCCA). In the results, we show the usefulness of the proposed method by predicting 1385 side-effects in the SIDER database from the chemical structures of 888 approved drugs. These predictions are performed with simultaneous extraction of correlated ensembles formed by a set of chemical substructures shared by drugs that are likely to have a set of side-effects. We also conduct a comprehensive side-effect prediction for many uncharacterized drug molecules stored in DrugBank, and were able to confirm interesting predictions using independent source of information.</p> <p>Conclusions</p> <p>The proposed method is expected to be useful in various stages of the drug development process.</p

Subgradient sampling for nonsmooth nonconvex minimization

Risk minimization for nonsmooth nonconvex problems naturally leads to firstorder sampling or, by an abuse of terminology, to stochastic subgradient descent. We establish the convergence of this method in the path-differentiable case, and describe more precise results under additional geometric assumptions. We recover and improve results from Ermoliev-Norkin by using a different approach: conservative calculus and the ODE method. In the definable case, we show that first-order subgradient sampling avoids artificial critical point with probability one and applies moreover to a large range of risk minimization problems in deep learning, based on the backpropagation oracle. As byproducts of our approach, we obtain several results on integration of independent interest, such as an interchange result for conservative derivatives and integrals, or the definability of set-valued parameterized integrals

Long term dynamics of the subgradient method for Lipschitz path differentiable functions

We consider the long-term dynamics of the vanishing stepsize subgradient method in the case when the objective function is neither smooth nor convex. We assume that this function is locally Lipschitz and path differentiable, i.e., admits a chain rule. Our study departs from other works in the sense that we focus on the behavior of the oscillations, and to do this we use closed measures. We recover known convergence results, establish new ones, and show a local principle of oscillation compensation for the velocities. Roughly speaking, the time average of gradients around one limit point vanishes. This allows us to further analyze the structure of oscillations, and establish their perpendicularity to the general drift

Long term dynamics of the subgradient method for Lipschitz path differentiable functions

We consider the long-term dynamics of the vanishing stepsize subgradient method in the case when the objective function is neither smooth nor convex. We assume that this function is locally Lipschitz and path differentiable, i.e., admits a chain rule. Our study departs from other works in the sense that we focus on the behavior of the oscillations, and to do this we use closed measures. We recover known convergence results, establish new ones, and show a local principle of oscillation compensation for the velocities. Roughly speaking, the time average of gradients around one limit point vanishes. This allows us to further analyze the structure of oscillations, and establish their perpendicularity to the general drift