In many applications, solutions of convex optimization problems are updated on-line, as functions of time. In this paper, we consider parametric semidefinite programs, which are linear optimization problems in the semidefinite cone whose coefficients (input data) depend on a time parameter. We are interested in the geometry of the solution (output data) trajectory, defined as the set of solutions depending on the parameter. We propose an exhaustive description of the geometry of the solution trajectory. As our main result, we show that only six distinct behaviors can be observed at a neighborhood of a given point along the solution trajectory. Each possible behavior is then illustrated by an example.
In this paper, we study parametric analysis of semidefinite optimization problems w.r.t. the perturbation of the objective function. We study the behavior of the optimal partition and optimal set mapping on a so-called nonlinearity interval. Furthermore, we investigate the sensitivity of the approximation of the optimal partition in a nonlinearity interval, which has been recently studied by Mohammad-Nezhad and Terlaky. The approximation of the optimal partition was obtained from a bounded sequence of interior solutions on, or in a neighborhood of the central path. We derive an upper bound on …
This paper revisits the parametric analysis of semidefinite optimization problems with respect to the perturbation of the objective function along a fixed direction. We review the notions of invariancy set, nonlinearity interval, and transition point of the optimal partition, and we investigate their characterizations. We show that the set of transition points is finite and the continuity of the optimal set mapping, on the basis of Painlevé-Kuratowski set convergence, might fail on a nonlinearity interval. Under a local nonsingularity condition, we then develop a methodology, stemming from num…
CNRS（法国国家科学研究中心）；LAAS（法国国家科学研究中心系统分析与架构实验室）；Université de Toulouse（图卢兹大学）；Czech Technical University in Prague（布拉格捷克理工大学）
We consider the problem of constructing an approximation of the Pareto curve associated with the multiobjective optimization problem $\min_{\mathbf{x} \in \mathbf{S}}\{ (f_1(\mathbf{x}), f_2(\mathbf{x})) \}$, where $f_1$ and $f_2$ are two conflicting polynomial criteria and $\mathbf{S} \subset \mathbb{R}^n$ is a compact basic semialgebraic set. We provide a systematic numerical scheme to approximate the Pareto curve. We start by reducing the initial problem into a scalarized polynomial optimization problem (POP). Three scalarization methods lead to consider different parametric POPs, namely (a…
We provide algorithms for computing a Lyapunov function for a class of systems where the state trajectories are constrained to evolve within a closed convex set. The dynamical systems that we consider comprise a differential equation which ensures continuous evolution within the domain, and a normal cone inclusion which ensures that the state trajectory remains within a prespecified set at all times. Finding a Lyapunov function for such a system boils down to finding a function which satisfies certain inequalities on the admissible set of state constraints. It is well-known that this problem, …
In this paper we develop a novel, discrete-time optimal control framework for mechanical systems with uncertain model parameters. We consider finite-horizon problems where the performance index depends on the statistical moments of the stochastic system. Our approach constitutes an extension of the original Differential Dynamic Programming method and handles uncertainty through generalized Polynomial Chaos (gPC) theory. The developed iterative scheme is capable of controlling the probabilistic evolution of the dynamic system and can be used in planning and control. Moreover, its scalable and f…
Given a compact parameter set $Y\subset R^p$, we consider polynomial optimization problems $(P_y$) on $R^n$ whose description depends on the parameter $y\inY$. We assume that one can compute all moments of some probability measure $ϕ$ on $Y$, absolutely continuous with respect to the Lebesgue measure (e.g. $Y$ is a box or a simplex and $ϕ$ is uniformly distributed). We then provide a hierarchy of semidefinite relaxations whose associated sequence of optimal solutions converges to the moment vector of a probability measure that encodes all information about all global optimal solutions $x^*(y)$…
Harvard University（哈佛大学）；University of California at Santa Barbara（加州大学圣巴巴拉分校）；Brown University（布朗大学）；University of Maryland at College Park（马里兰大学学院公园分校）
We consider semidefinite programming (SDP) approaches for solving the maximum satisfiability problem (MAX-SAT) and the weighted partial MAX-SAT. It is widely known that SDP is well-suited to approximate the (MAX-)2-SAT. Our work shows the potential of SDP also for other satisfiability problems, by being competitive with some of the best solvers in the yearly MAX-SAT competition. Our solver combines sum of squares (SOS) based SDP bounds and an efficient parser within a branch & bound scheme. On the theoretical side, we propose a family of semidefinite feasibility problems, and show that a membe…
We consider a parametric convex quadratic programming, CQP, relaxation for the quadratic knapsack problem, QKP. This relaxation maintains partial quadratic information from the original QKP by perturbing the objective function to obtain a concave quadratic term. The nonconcave part generated by the perturbation is then linearized by a standard approach that lifts the problem to the matrix space. We present a primal-dual interior point method to optimize the perturbation of the quadratic function, in a search for the tightest upper bound for the \QKPp. We prove that the same perturbation approa…
Semidefinite programming can be considered over any real closed field, including fields of Puiseux series equipped with their nonarchimedean valuation. Nonarchimedean semidefinite programs encode parametric families of classical semidefinite programs, for sufficiently large values of the parameter. Recently, a correspondence has been established between nonarchimedean semidefinite programs and stochastic mean payoff games with perfect information. This correspondence relies on tropical geometry. It allows one to solve generic nonarchimedean semidefinite feasibility problems, of large scale, by…
This paper studies convex Generalized Nash Equilibrium Problems (GNEPs) that are given by polynomials. We use rational and parametric expressions for Lagrange multipliers to formulate efficient polynomial optimization for computing Generalized Nash Equilibria (GNEs). The Moment-SOS hierarchy of semidefinite relaxations are used to solve the polynomial optimization. Under some general assumptions, we prove the method can find a GNE if there exists one, or detect nonexistence of GNEs. Numerical experiments are presented to show the efficiency of the method.
A spectrahedron is the feasible set of a semidefinite program, SDP, i.e., the intersection of an affine set with the positive semidefinite cone. While strict feasibility is a generic property for random problems, there are many classes of problems where strict feasibility fails and this means that strong duality can fail as well. If the minimal face containing the spectrahedron is known, the SDPcan easily be transformed into an equivalent problem where strict feasibility holds and thus strong duality follows as well. The minimal face is fully characterized by the range or nullspace of any of t…
Complete Trajectory Tracking for Polynomial Differential Variational Inequalities: A Moment-SOS Based Structure-Aware Approach
School of Mathematical Sciences, Chengdu University of Technology（成都理工大学数学科学学院）；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数学科学重点实验室）
In this paper, we address the problem of closed-loop control of nonlinear dynamical systems subjected to probabilistic uncertainties. More precisely, we design time-varying polynomial feedback controllers to follow the given nominal trajectory and also, for safety purposes, remain in the tube around the nominal trajectory, despite all uncertainties. We formulate this problem as a chance optimization problem where we maximize the probability of achieving control objectives. To address control problems with long planning horizons, we formulate the single large chance optimization problem as a se…
We present a new algorithm for solving a polynomial program P based on the recent "joint + marginal" approach of the first author for, parametric optimization. The idea is to first consider the variable x1 as a parameter and solve the associated (n-1)-variable (x2,...,xn) problem P(x1) where the parameter x1 is fixed and takes values in some interval Y1 with some probability uniformly distributed on Y1. Then one considers the hierarchy of what we call "joint+marginal" semidefinite relaxations, whose duals provide a sequence of univariate polynomial approximations that converges to the optimal …
In this two-part study we develop a unified approach to the analysis of the global exactness of various penalty and augmented Lagrangian functions for finite-dimensional constrained optimization problems. This approach allows one to verify in a simple and straightforward manner whether a given penalty/augmented Lagrangian function is exact, i.e. whether the problem of unconstrained minimization of this function is equivalent (in some sense) to the original constrained problem, provided the penalty parameter is sufficiently large. Our approach is based on the so-called localization principle th…
Mixtures Closest to a Given Measure: A Semidefinite Programming Approach
College of Computing and Data Science, NTU, Singapore（新加坡国立大学计算与数据科学学院）；LAAS-CNRS, Toulouse, France（法国图卢兹CNRS-Laas研究所）；Toulouse School of Economics（图卢兹经济学院）
Control of nonlinear dynamical systems is a complex and multifaceted process. Essential elements of many engineering systems include high fidelity physics-based modeling, offline trajectory planning, feedback control design, and data acquisition strategies to reduce uncertainties. This article proposes an optimization centric perspective which couples these elements in a cohesive framework. We introduce a novel use of hyper-differential sensitivity analysis to understand the sensitivity of feedback controllers to parametric uncertainty in physics-based models used for trajectory planning. Thes…
Generalized Polynomial Chaos (gPC) theory has been widely used for representing parametric uncertainty in a system, thanks to its ability to propagate uncertainty evolution. In an optimal control context, gPC can be combined with several optimization techniques to achieve a control policy that handles effectively this type of uncertainty. Such a suitable method is Differential Dynamic Programming (DDP), leading to an algorithm that inherits the scalability to high-dimensional systems and fast convergence nature of the latter. In this paper, we expand this combination aiming to acquire probabil…
This paper considers a two-dimensional persistent monitoring problem by controlling movements of second-order agents to minimize some uncertainty metric associated with targets in a dynamic environment. In contrast to common sensing models depending only on the distance from a target, we introduce an active sensing model which considers the distance between an agent and a target, as well as the agent's velocity. We propose an objective function which can achieve a collision-free agent trajectory by penalizing all possible collisions. Applying structural properties of the optimal control derive…
We describe methods for proving bounds on infinite-time averages in differential dynamical systems. The methods rely on the construction of nonnegative polynomials with certain properties, similarly to the way nonlinear stability can be proved using Lyapunov functions. Nonnegativity is enforced by requiring the polynomials to be sums of squares, a condition which is then formulated as a semidefinite program (SDP) that can be solved computationally. Although such computations are subject to numerical error, we demonstrate two ways to obtain rigorous results: using interval arithmetic to control…
Many problems of systems control theory boil down to solving polynomial equations, polynomial inequalities or polyomial differential equations. Recent advances in convex optimization and real algebraic geometry can be combined to generate approximate solutions in floating point arithmetic. In the first part of the course we describe semidefinite programming (SDP) as an extension of linear programming (LP) to the cone of positive semidefinite matrices. We investigate the geometry of spectrahedra, convex sets defined by linear matrix inequalities (LMIs) or affine sections of the SDP cone. We als…
We consider the problem of optimal trajectory tracking for unknown systems. A novel data-enabled predictive control (DeePC) algorithm is presented that computes optimal and safe control policies using real-time feedback driving the unknown system along a desired trajectory while satisfying system constraints. Using a finite number of data samples from the unknown system, our proposed algorithm uses a behavioural systems theory approach to learn a non-parametric system model used to predict future trajectories. The DeePC algorithm is shown to be equivalent to the classical and widely adopted Mo…
In this article we present a generalised view on Path Integral Control (PIC) methods. PIC refers to a particular class of policy search methods that are closely tied to the setting of Linearly Solvable Optimal Control (LSOC), a restricted subclass of nonlinear Stochastic Optimal Control (SOC) problems. This class is unique in the sense that it can be solved explicitly to yield a formal optimal state trajectory distribution. In this contribution we first review the PIC theory and discuss related algorithms tailored to policy search in general. We are able to identify a generic design strategy t…
An adaptive approach to using reduced-order models as surrogates in PDE-constrained optimization is introduced that breaks the traditional offline-online framework of model order reduction. A sequence of optimization problems constrained by a given Reduced-Order Model (ROM) is defined with the goal of converging to the solution of a given PDE-constrained optimization problem. For each reduced optimization problem, the constraining ROM is trained from sampling the High-Dimensional Model (HDM) at the solution of some of the previous problems in the sequence. The reduced optimization problems are…
We present a mathematical and computational framework for the problem of learning a dynamical system from noisy observations of a few trajectories and subject to side information. Side information is any knowledge we might have about the dynamical system we would like to learn besides trajectory data. It is typically inferred from domain-specific knowledge or basic principles of a scientific discipline. We are interested in explicitly integrating side information into the learning process in order to compensate for scarcity of trajectory observations. We identify six types of side information …
