We present a method for the approximate propagation of mean and covariance of a probability distribution through ordinary differential equations (ODE) with discontinous right-hand side. For piecewise affine systems, a normalization of the propagated probability distribution at every time step allows us to analytically compute the expectation integrals of the mean and covariance dynamics while explicitly taking into account the discontinuity. This leads to a natural smoothing of the discontinuity such that for relevant levels of uncertainty the resulting ODE can be integrated directly with standard schemes and it is neither necessary to prespecify the switching sequence nor to use a switch detection method. We then show how this result can be employed in the more general case of piecewise smooth functions based on a structure preserving linearization scheme. The resulting dynamics can be straightforwardly used within standard formulations of stochastic optimal control problems with chance constraints.
In this paper, we present some new necessary and sufficient optimality conditions in terms of the Clarke subdifferentials for approximate Pareto solutions of a nonsmooth vector optimization problem which has an infinite number of constraints. As a consequence, we obtain optimality conditions for the particular cases of cone-constrained convex vector optimization problems and semidefinite vector optimization problems. Examples are given to illustrate the obtained results.
This paper deals with approximate solutions of a nonsmooth semi-infinite programming with multiple interval-valued objective functions. We first introduce four types of approximate quasi Pareto solutions of the considered problem by considering the lower-upper interval order relation and then apply some advanced tools of variational analysis and generalized differentiation to establish necessary optimality conditions for these approximate solutions. Sufficient conditions for approximate quasi Pareto solutions of such a problem are also provided by means of introducing the concepts of approxima…
An Approximate Conjugate Subgradient Algorithm with Matrix Parameter for Derivative-Free Nonsmooth Optimization Problems
Fakultät für Mathematik, Universität Wien（维也纳大学）；Faculty of Engineering, Free University of Bozen–Bolzano（博岑-博尔扎诺自由大学）；Université Catholique de Louvain (UCLouvain)（天主教鲁汶大学）
A class of exact penalty-type local search methods for optimal control problems with nonsmooth cost functional, nonsmooth (but continuous) dynamics, and nonsmooth state and control constraints is presented, in which the the penalty parameter and several line search parameters are adaptively adjusted during the optimisation process. This class of methods is applicable to problems having a known DC (Difference-of-Convex functions) structure and in its core is based on the classical DCA method, combined with the steering exact penalty rules for updating the penalty parameter and an adaptive nonmo…
We study unconstrained optimization problems with nonsmooth and convex objective function in the form of a mathematical expectation. The proposed method approximates the expected objective function with a sample average function using Inexact Restoration-based adapted sample sizes. The sample size is chosen in an adaptive manner based on Inexact Restoration. The algorithm uses line search and assumes descent directions with respect to the current approximate function. We prove the a.s. convergence under standard assumptions. Numerical results for two types of problems, machine learning loss fu…
In this paper we study the optimal control of a class of semilinear elliptic partial differential equations which have nonlinear constituents that are only accessible by data and are approximated by nonsmooth ReLU neural networks. The optimal control problem is studied in detail. In particular, the existence and uniqueness of the state equation are shown, and continuity as well as directional differentiability properties of the corresponding control-to-state map are established. Based on approximation capabilities of the pertinent networks, we address fundamental questions regarding approximat…
We propose and analyze a numerical algorithm for solving a class of optimal control problems for learning-informed semilinear partial differential equations. The latter is a class of PDEs with constituents that are in principle unknown and are approximated by nonsmooth ReLU neural networks. We first show that a direct smoothing of the ReLU network with the aim to make use of classical numerical solvers can have certain disadvantages, namely potentially introducing multiple solutions for the corresponding state equation. This motivates us to devise a numerical algorithm that treats directly the…
We introduce the concept of strong high-order approximate minimizers for nonconvex optimization problems. These apply in both standard smooth and composite non-smooth settings, and additionally allow convex or inexpensive constraints. An adaptive regularization algorithm is then proposed to find such approximate minimizers. Under suitable Lipschitz continuity assumptions, whenever the feasible set is convex, it is shown that using a model of degree $p$, this algorithm will find a strong approximate q-th-order minimizer in at most ${\cal O}\left(\max_{1\leq j\leq q}ε_j^{-(p+1)/(p-j+1)}\right)$ …
Zeroth-Order Methods for Stochastic Nonconvex Nonsmooth Composite Optimization
Department of Computer Science University of Maryland College Park（计算机科学系大学马里兰大学学院公园分校）；Department of Computer Science and Engineering University of Texas Arlington（计算机科学与工程系德克萨斯大学阿灵顿分校）
A successive difference-of-convex method for a class of two-stage nonconvex nonsmooth stochastic conic program via SVI
School of Mathematics and Statistics, Beijing Jiaotong University（北京交通大学数学与统计学院）；Beijing Key Laboratory of Biological Big Data and Topological Statistics, Beijing Jiaotong University（北京交通大学生物大数据与拓扑统计北京市重点实验室）
Optimization problems with uncertainty in the constraints occur in many applications. Particularly, probability functions present a natural form to deal with this situation. Nevertheless, in some cases, the resulting probability functions are nonsmooth. This motivates us to propose a regularization employing the Moreau envelope of a scalar representation of the vector inequality. More precisely, we consider a probability function which covers most of the general classes of probabilistic constraints: $$φ(x)=\mathbb{P}(Φ(x,ξ)\in -\mathcal{K}),$$ where $\mathcal{K}$ is a convex cone of a Banach s…
A Stochastic Successive Minimization Method for Nonsmooth Nonconvex Optimization with Applications to Transceiver Design in Wireless Communication Networks
In this paper, a novel stochastic extra-step quasi-Newton method is developed to solve a class of nonsmooth nonconvex composite optimization problems. We assume that the gradient of the smooth part of the objective function can only be approximated by stochastic oracles. The proposed method combines general stochastic higher order steps derived from an underlying proximal type fixed-point equation with additional stochastic proximal gradient steps to guarantee convergence. Based on suitable bounds on the step sizes, we establish global convergence to stationary points in expectation and an ext…
CNRS UMR 6004, LS2N, École Centrale de Nantes（法国国家科学研究中心 UMR 6004，南特数字科学实验室，南特中央理工大学）；IRIT, Université de Toulouse, CNRS（图卢兹计算机科学研究所，图卢兹大学，法国国家科学研究中心）
We consider flows of ordinary differential equations (ODEs) driven by path differentiable vector fields. Path differentiable functions constitute a proper subclass of Lipschitz functions which admit conservative gradients, a notion of generalized derivative compatible with basic calculus rules. Our main result states that such flows inherit the path differentiability property of the driving vector field. We show indeed that forward propagation of derivatives given by the sensitivity differential inclusions provide a conservative Jacobian for the flow. This allows to propose a nonsmooth version…
In this paper, we establish that for a wide class of controlled stochastic differential equations (SDEs) with stiff coefficients, the value functions of corresponding zero-sum games can be represented by a deep artificial neural network (DNN), whose complexity grows at most polynomially in both the dimension of the state equation and the reciprocal of the required accuracy. Such nonlinear stiff systems may arise, for example, from Galerkin approximations of controlled stochastic partial differential equations (SPDEs), or controlled PDEs with uncertain initial conditions and source terms. This …
Optimal control problems with free terminal time present many challenges including nonsmooth and discontinuous control laws, irregular value functions, many local optima, and the curse of dimensionality. To overcome these issues, we propose an adaptation of the model-based actor-critic paradigm from the field of Reinforcement Learning via an exponential transformation to learn an approximate feedback control and value function pair. We demonstrate the algorithm's effectiveness on prototypical examples featuring each of the main pathological issues present in problems of this type.
In this paper we study second order master equations arising from mean field games with common noise over arbitrary time duration. A classical solution typically requires the monotonicity condition (or small time duration) and sufficiently smooth data. While keeping the monotonicity condition, our goal is to relax the regularity of the data, which is an open problem in the literature. In particular, we do not require any differentiability in terms of the measures, which prevents us from obtaining classical solutions. We shall propose three weaker notions of solutions, named as {\it good soluti…
We consider two greedy algorithms for minimizing a convex function in a bounded convex set: an algorithm by Jones [1992] and the Frank-Wolfe (FW) algorithm. We first consider approximate versions of these algorithms. For smooth convex functions, we give sufficient conditions for convergence, a unified analysis for the well-known convergence rate of O(1/k) together with a result showing that this rate is the best obtainable from the proof technique, and an equivalence result for the two algorithms. We also consider approximate stochastic greedy algorithms for minimizing expectations. We show th…
We propose and analyze several stochastic gradient algorithms for finding stationary points or local minimum in nonconvex, possibly with nonsmooth regularizer, finite-sum and online optimization problems. First, we propose a simple proximal stochastic gradient algorithm based on variance reduction called ProxSVRG+. We provide a clean and tight analysis of ProxSVRG+, which shows that it outperforms the deterministic proximal gradient descent (ProxGD) for a wide range of minibatch sizes, hence solves an open problem proposed in Reddi et al. (2016b). Also, ProxSVRG+ uses much less proximal oracle…
Empirical risk minimization for risk-neutral composite optimal control with applications to bang-bang control
H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院 H. 米尔顿·斯图尔特工业与系统工程学院）；Institut für Mathematik, Humboldt-Universität zu Berlin（柏林洪堡大学数学研究所）
