In model predictive control (MPC) an optimization problem has to be solved at each time step, which in real-time applications makes it important to solve these optimization problems efficiently and to have good upper bounds on worst-case solution time. Often for linear MPC problems, the optimization problem in question is a quadratic program (QP) that depends on parameters such as system states and reference signals. A popular class of methods for solving such QPs is active-set methods, where a sequence of linear systems of equations is solved. We propose an algorithm for computing which sequence of subproblems an active-set algorithm will solve, for every parameter of interest. By knowing these sequences, a worst-case bound on how many iterations, and ultimately the maximum time, the active-set algorithm requires to converge can be determined. The usefulness of the proposed method is illustrated on a set of QPs, originating from MPC problems, by computing the exact worst-case number of iterations primal and dual active-set algorithms require to reach optimality.
In this chapter we derive computational complexity certifications of first order inexact dual methods for solving general smooth constrained convex problems which can arise in real-time applications, such as model predictive control. When it is difficult to project on the primal constraint set described by a collection of general convex functions, we use the Lagrangian relaxation to handle the complicated constraints and then, we apply dual (fast) gradient algorithms based on inexact dual gradient information for solving the corresponding dual problem. The iteration complexity analysis is base…
InFlow: entropic stochastic dual dynamic programming with HJB cross-certification for seasonal energy storage
Cheikh Anta Diop University of Dakar（达喀尔谢赫安塔·迪奥普大学）；University of Nairobi（内罗毕大学）；Laboratoire Eau, Energie, Environnement et procédés Industriels (LE3PI)（水、能源、环境与工业过程实验室）；Centre de Tests des Systèmes Solaires (CT2S)（太阳能系统测试中心）
For paving the way to novel applications in quantum simulation, computation, and technology, increasingly large quantum systems have to be steered with high precision. It is a typical task amenable to numerical optimal control to turn the time course of pulses, i.e. piecewise constant control amplitudes, iteratively into an optimised shape. Here, we present the first comparative study of optimal control algorithms for a wide range of finite-dimensional applications. We focus on the most commonly used algorithms: GRAPE methods which update all controls concurrently, and KROTOV-type methods whic…
We provide a lower bound showing that the $O(1/k)$ convergence rate of the NoLips method (a.k.a. Bregman Gradient) is optimal for the class of functions satisfying the $h$-smoothness assumption. This assumption, also known as relative smoothness, appeared in the recent developments around the Bregman Gradient method, where acceleration remained an open issue. On the way, we show how to constructively obtain the corresponding worst-case functions by extending the computer-assisted performance estimation framework of Drori and Teboulle (Mathematical Programming, 2014) to Bregman first-order meth…
University of California, Berkeley（加州大学伯克利分校）；Tsinghua-Berkeley Shenzhen Institute（清华-伯克利深圳学院）
In this paper, we consider the problem of certifying the robustness of neural networks to perturbed and adversarial input data. Such certification is imperative for the application of neural networks in safety-critical decision-making and control systems. Certification techniques using convex optimization have been proposed, but they often suffer from relaxation errors that void the certificate. Our work exploits the structure of ReLU networks to improve relaxation errors through a novel partition-based certification procedure. The proposed method is proven to tighten existing linear programmi…
Quantum information leverages properties of quantum behaviors in order to perform useful tasks such as secure communication and randomness certification. Nevertheless, not much is known about the intricate geometric features of the set quantum behaviors. In this paper we study the structure of the set of quantum correlators using semidefinite programming. Our main results are (i) a generalization of the analytic description by Tsirelson-Landau-Masanes, (ii) necessary and sufficient conditions for extremality and exposedness, and (iii) an operational interpretation of extremality in the case of…
We consider challenging dynamic programming models where the associated Bellman equation, and the value and policy iteration algorithms commonly exhibit complex and even pathological behavior. Our analysis is based on the new notion of regular policies. These are policies that are well-behaved with respect to value and policy iteration, and are patterned after proper policies, which are central in the theory of stochastic shortest path problems. We show that the optimal cost function over regular policies may have favorable value and policy iteration properties, which the optimal cost function…
We present a unifying framework for generating extended formulations for the polyhedral outer approximations used in algorithms for mixed-integer convex programming (MICP). Extended formulations lead to fewer iterations of outer approximation algorithms and generally faster solution times. First, we observe that all MICP instances from the MINLPLIB2 benchmark library are conic representable with standard symmetric and nonsymmetric cones. Conic reformulations are shown to be effective extended formulations themselves because they encode separability structure. For mixed-integer conic-representa…
Many algorithms use data structures that maintain properties of matrices undergoing some changes. The applications are wide-ranging and include for example matchings, shortest paths, linear programming, semi-definite programming, convex hull and volume computation. Given the wide range of applications, the exact property these data structures must maintain varies from one application to another, forcing algorithm designers to invent them from scratch or modify existing ones. Thus it is not surprising that these data structures and their proofs are usually tailor-made for their specific applica…
Scalable Ground-State Certification of Quantum Spin Systems via Structured Noncommutative Polynomial Optimization
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology（巴塞罗那科技研究所光科学中心）；Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-PSL Research University, Collège de France（索邦大学卡斯特勒-布罗塞尔实验室）；Inria Paris-Saclay（巴黎萨克雷国家信息与自动化研究所）；LAAS-CNRS & Institute of Mathematics from Toulouse（图卢兹数学研究所）；ICREA - Institució Catalana de Recerca i Estudis Avançats（加泰罗尼亚研究与高等教育局）
Information propagation on networks is a central theme in social, behavioral, and economic sciences, with important theoretical and practical implications, such as the influence maximization problem for viral marketing. Here, we consider a model that unifies the classical independent cascade models and the linear threshold models, and generalise them by considering continuous variables and allowing feedback in the dynamics. We then formulate its influence maximization as a mixed integer nonlinear programming problem and adopt derivative-free methods. Furthermore, we show that the problem can b…
LAAS-CNRS（法国国家科学研究中心自动化、计算机工程、信号与机器人实验室）；Université de Toulouse（图卢兹大学）；IMT（图卢兹理工学院）；IRIT（图卢兹计算机研究所）
The Lipschitz constant of a network plays an important role in many applications of deep learning, such as robustness certification and Wasserstein Generative Adversarial Network. We introduce a semidefinite programming hierarchy to estimate the global and local Lipschitz constant of a multiple layer deep neural network. The novelty is to combine a polynomial lifting for ReLU functions derivatives with a weak generalization of Putinar's positivity certificate. This idea could also apply to other, nearly sparse, polynomial optimization problems in machine learning. We empirically demonstrate th…
Verifying that input-output relationships of a neural network conform to prescribed operational specifications is a key enabler towards deploying these networks in safety-critical applications. Semidefinite programming (SDP)-based approaches to Rectified Linear Unit (ReLU) network verification transcribe this problem into an optimization problem, where the accuracy of any such formulation reflects the level of fidelity in how the neural network computation is represented, as well as the relaxations of intractable constraints. While the literature contains much progress on improving the tightne…
Université Paris-Saclay（巴黎萨克雷大学）；Centre National de la Recherche Scientifique (CNRS)（法国国家科学研究中心）；CentraleSupélec（巴黎中央理工-高等电力学院）；Stanford University（斯坦福大学）
Sequential Convex Programming (SCP) has recently gained significant popularity as an effective method for solving optimal control problems and has been successfully applied in several different domains. However, the theoretical analysis of SCP has received comparatively limited attention, and it is often restricted to discrete-time formulations. In this paper, we present a unifying theoretical analysis of a fairly general class of SCP procedures for continuous-time optimal control problems. In addition to the derivation of convergence guarantees in a continuous-time setting, our analysis revea…
We consider the problem of verifying safety for a pair of identical integrator agents in continuous time with compact set-valued input uncertainties. We encode this verification problem as that of certifying or falsifying the intersection of their reach sets. We transcribe the same into a variational problem, namely that of minimizing the support function of the difference of the two reach sets over the unit sphere. We illustrate the computational tractability of the proposed formulation by developing two cases in detail, viz. when the inputs have time-varying norm-bounded and generic hyperrec…
Purdue University（普渡大学）；University of Melbourne（墨尔本大学）；Massachusetts Institute of Technology（麻省理工学院）；UC Berkeley（加州大学伯克利分校）；Rice University（莱斯大学）；Carnegie Mellon University（卡内基梅隆大学）
Semidefinite programming (SDP) is a unifying framework that generalizes both linear programming and quadratically-constrained quadratic programming, while also yielding efficient solvers, both in theory and in practice. However, there exist known impossibility results for approximating the optimal solution when constraints for covering SDPs arrive in an online fashion. In this paper, we study online covering linear and semidefinite programs in which the algorithm is augmented with advice from a possibly erroneous predictor. We show that if the predictor is accurate, we can efficiently bypass t…
