Some applications of polynomial optimization in operations research and real-time decision making

Abstract-Exciting recent developments at the interface of optimization and control have shown that several fundamental problems in dynamics and control, such as stability, collision avoidance, robust performance, and controller synthesis can be addressed by a synergy of classical tools from Lyapunov theory and modern computational techniques from algebraic optimization. In this paper, we give a brief overview of our recent research efforts (with various coauthors) to (i) enhance the scalability of the algorithms in this field, and (ii) understand their worst case performance guarantees as well as fundamental limitations. Our results are tersely surveyed and challenges/opportunities that lie ahead are stated. I. ALGEBRAIC METHODS IN OPTIMIZATION AND CONTROLIn recent years, a fundamental and exciting interplay between convex optimization and algorithmic algebra has allowed for the solution or approximation of a large class of nonlinear and nonconvex problems in optimization and control once thought to be out of reach. The success of this area stems from two facts: (i) Numerous fundamental problems in optimization and control (among several other disciplines in applied and computational mathematics) are semialgebraic; i.e., they involve optimization over sets defined by a finite number of possibly quantified polynomial inequalities. (ii) Semialgebraic problems can be reformulated as optimization problems over the set of nonnegative polynomials. This makes them amenable to a rich set of algebraic tools which lend themselves to semidefinite programming-a subclass of convex optimization problems for which global solution methods are available.Application areas within optimization and computational mathematics that have been impacted by advances in algebraic techniques are numerous: approximation algorithms for NP-hard combinatorial problems from classical linear control to a principled framework for design of nonlinear (polynomial) controllers that are provably safer, more agile, and more robust. As a concrete example, Figure 1 demonstrates our recent work with Majumdar and Tedrake [14] in this area applied to the field of robotics. As the caption explains, sos techniques provide contollers with much larger margins of safety along planned trajectories and can directly reason about the nonlinear dynamics of the system under consideration. These are crucial assets for more challenging robotic tasks such as walking, running, and flying. Sum of squares methods have also recently made their way to actual industry flight control problems, e.g., to explain the falling leaf mode phenomenon of the F/A-18 Hornet aircraft [15], [16] or to design controllers for hypersonic aircraft [17]. The "swing-up and balance" task via sum of squares optimization for an underactuated and severely torque limited double pendulum (the "Acrobot"). Top: projections of basins of attraction around a nominal swing-up trajectory designed by linear quadratic regulator (LQR) techniques (blue) and by SOS techniques (red). Bottom: projections of basins of...

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“…A large portion of our recent papers [27], [28], [29] is devoted to this question. We provide a glimpse of the results in this section.…”

Section: B How Fast/powerful Is the Dsos And Sdsos Methodology?mentioning

confidence: 99%

Towards scalable algorithms with formal guarantees for Lyapunov analysis of control systems via algebraic optimization

Ahmadi

Parrilo²

2014

53rd IEEE Conference on Decision and Control

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“…A nonlinear control problem of this scale is beyond the reach of standard solvers for sum of squares programs at the moment. In a different paper [9], the authors show the potential of DSOS and SDSOS optimization for real-time applications. More specifically, they use these techniques to compute, every few milliseconds, certificates of collision avoidance for a simple model of an unmanned vehicle that navigates through a cluttered environment.…”

Section: Dsos and Sdsos Optimizationmentioning

confidence: 99%

Recent Scalability Improvements for Semidefinite Programming with Applications in Machine Learning, Control, and Robotics

Majumdar

Hall

Ahmadi

2020

Annu. Rev. Control Robot. Auton. Syst.

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Historically, scalability has been a major challenge to the successful application of semidefinite programming in fields such as machine learning, control, and robotics. In this paper, we survey recent approaches for addressing this challenge including (i) approaches for exploiting structure (e.g., sparsity and symmetry) in a problem, (ii) approaches that produce low-rank approximate solutions to semidefinite programs, (iii) more scalable algorithms that rely on augmented Lagrangian techniques and the alternating direction method of multipliers, and (iv) approaches that trade off scalability with conservatism (e.g., by approximating semidefinite programs with linear and second-order cone programs). For each class of approaches we provide a high-level exposition, an entry-point to the corresponding literature, and examples drawn from machine learning, control, or robotics. We also present a list of software packages that implement many of the techniques discussed in the paper. Our hope is that this paper will serve as a gateway to the rich and exciting literature on scalable semidefinite programming for both theorists and practitioners.

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“…The computational advantage of SOS programming stems from its intrinsic link to SDPs. Specifically, a polynomial p of degree 2d is SOS if and only if p(x) = z(x) T Qz(x), where Q 0 and z(x) is a vector of monomials up to order d. Thus, certifying that a polynomial is SOS reduces to the task of finding a psd matrix Q subject to a finite set of linear equalities, thus taking the form in (2). Certificates of the form in (4) will form the building block for our approach.…”

Section: Sum-of-squares (Sos) Programming Backgroundmentioning

confidence: 99%

Robust Tracking with Model Mismatch for Fast and Safe Planning: an SOS Optimization Approach

Singh¹,

Chen²,

Herbert³

et al. 2019

EasyChair Preprints

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In the pursuit of real-time motion planning, a commonly adopted practice is to compute trajectories by running a planning algorithm on a simplified, low-dimensional dynamical model, and then employ a feedback tracking controller that tracks such a trajectory by accounting for the full, high-dimensional system dynamics. While this strategy of planning with model mismatch generally yields fast computation times, there are no guarantees of dynamic feasibility, which hampers application to safety-critical systems. Building upon recent work that addressed this problem through the lens of Hamilton-Jacobi (HJ) reachability, we devise an algorithmic framework whereby one computes, offline, for a pair of "planner" (i.e., low-dimensional) and "tracking" (i.e., high-dimensional) models, a feedback tracking controller and associated tracking bound. This bound is then used as a safety margin when generating motion plans via the low-dimensional model. Specifically, we harness the computational tool of sum-of-squares (SOS) programming to design a bilinear optimization algorithm for the computation of the feedback tracking controller and associated tracking bound. The algorithm is demonstrated via numerical experiments, with an emphasis on investigating the trade-off between the increased computational scalability afforded by SOS and its intrinsic conservativeness. Collectively, our results enable scaling the appealing strategy of planning with model mismatch to systems that are beyond the reach of HJ analysis, while maintaining safety guarantees.

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Some applications of polynomial optimization in operations research and real-time decision making

Cited by 46 publications

References 33 publications

Towards scalable algorithms with formal guarantees for Lyapunov analysis of control systems via algebraic optimization

Towards scalable algorithms with formal guarantees for Lyapunov analysis of control systems via algebraic optimization

Recent Scalability Improvements for Semidefinite Programming with Applications in Machine Learning, Control, and Robotics

Robust Tracking with Model Mismatch for Fast and Safe Planning: an SOS Optimization Approach

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