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Benign Nonconvex Landscapes in Optimal and Robust Control, Part II: Extended Convex Lifting

Yang Zheng, Chih-Fan Pai, Yujie Tang

TL;DR

The paper introduces Extended Convex Lifting (ECL), a unifying framework that reveals hidden convexity in benign nonconvex control problems by lifting epigraphs to higher dimensions and mapping them via smooth diffeomorphisms to convex (often LMI-representable) sets. It provides theoretical guarantees that minimizing the original nonconvex objective is equivalent to a convex program and that any nondegenerate Clarke stationary point is globally optimal. The framework is instantiated for key problems including state-feedback LQR, state-feedback and output-feedback $\mathcal{H}_\infty$, LQG, and distributed control under Quadratic Invariance, yielding new global optimality results (notably for LQG and H∞ output feedback) and unifying prior LMI-based approaches. The results advance both theory and practice by enabling convex analyses and model-free insights for a broad class of nonconvex policy optimization problems, potentially extending beyond control to other domains with similar structures.

Abstract

Many optimal and robust control problems are nonconvex and potentially nonsmooth in their policy optimization forms. In Part II of this paper, we introduce a new and unified Extended Convex Lifting (ECL) framework to reveal hidden convexity in classical optimal and robust control problems from a modern optimization perspective. Our ECL offers a bridge between nonconvex policy optimization and convex reformulations, enabling convex analysis for nonconvex problems. Despite non-convexity and non-smoothness, the existence of an ECL not only reveals that minimizing the original function is equivalent to a convex problem but also certifies a class of first-order non-degenerate stationary points to be globally optimal. Therefore, no spurious stationarity exists in the set of non-degenerate policies. This ECL framework can cover many benchmark control problems, including state feedback linear quadratic regulator (LQR), dynamic output feedback linear quadratic Gaussian (LQG) control, and $\mathcal{H}_\infty$ robust control. ECL can also handle a class of distributed control problems when the notion of quadratic invariance (QI) holds. We further show that all static stabilizing policies are non-degenerate for state feedback LQR and $\mathcal{H}_\infty$ control under standard assumptions. We believe that the new ECL framework may be of independent interest for analyzing nonconvex problems beyond control.

Benign Nonconvex Landscapes in Optimal and Robust Control, Part II: Extended Convex Lifting

TL;DR

The paper introduces Extended Convex Lifting (ECL), a unifying framework that reveals hidden convexity in benign nonconvex control problems by lifting epigraphs to higher dimensions and mapping them via smooth diffeomorphisms to convex (often LMI-representable) sets. It provides theoretical guarantees that minimizing the original nonconvex objective is equivalent to a convex program and that any nondegenerate Clarke stationary point is globally optimal. The framework is instantiated for key problems including state-feedback LQR, state-feedback and output-feedback , LQG, and distributed control under Quadratic Invariance, yielding new global optimality results (notably for LQG and H∞ output feedback) and unifying prior LMI-based approaches. The results advance both theory and practice by enabling convex analyses and model-free insights for a broad class of nonconvex policy optimization problems, potentially extending beyond control to other domains with similar structures.

Abstract

Many optimal and robust control problems are nonconvex and potentially nonsmooth in their policy optimization forms. In Part II of this paper, we introduce a new and unified Extended Convex Lifting (ECL) framework to reveal hidden convexity in classical optimal and robust control problems from a modern optimization perspective. Our ECL offers a bridge between nonconvex policy optimization and convex reformulations, enabling convex analysis for nonconvex problems. Despite non-convexity and non-smoothness, the existence of an ECL not only reveals that minimizing the original function is equivalent to a convex problem but also certifies a class of first-order non-degenerate stationary points to be globally optimal. Therefore, no spurious stationarity exists in the set of non-degenerate policies. This ECL framework can cover many benchmark control problems, including state feedback linear quadratic regulator (LQR), dynamic output feedback linear quadratic Gaussian (LQG) control, and robust control. ECL can also handle a class of distributed control problems when the notion of quadratic invariance (QI) holds. We further show that all static stabilizing policies are non-degenerate for state feedback LQR and control under standard assumptions. We believe that the new ECL framework may be of independent interest for analyzing nonconvex problems beyond control.
Paper Structure (50 sections, 29 theorems, 246 equations, 9 figures)

This paper contains 50 sections, 29 theorems, 246 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Non-convexity and Non-smoothness in Control
  3. Our Techniques --- Extended Convex Lifting ($\mathtt{ECL}$)
  4. Related Work
  5. Solutions of optimal LQG control and $\mathcal{H}_\infty$ robust control.
  6. Lifting strategies in analyzing nonconvex problems
  7. Paper Outline
  8. Notations.
  9. Motivating Examples
  10. Three Examples
  11. A Simple Fact on Epigraphs
  12. Extended Convex Lifting ($\mathtt{ECL}$) for Benign Non-convexity
  13. A Framework of Extended Convex Lifting (ECL)
  14. Non-degenerate Stationarity Implies Global Optimality
  15. Proof of \ref{['theorem:ECL-guarantee']}
  16. ...and 35 more sections

Key Result

Theorem 3.1

Let $f:\mathcal{D}\rightarrow\mathbb{R}$ be a continuous function equipped with an $\mathtt{ECL}$$(\mathcal{L}_{\mathrm{lft}},\mathcal{F}_{\mathrm{cvx}},\mathcal{G}_{\mathrm{aux}},\Phi)$. Then, we have

Figures (9)

  • Figure 1: Illustration of nonconvexity and non-smoothness in control: (a) a slice of the set of static feedback gains $K$ such that the closed-loop system $A+BK$ is stable, where $A = 0, B = I_2$; (b) nonconvex and smooth LQR cost; (c) nonsmooth and nonconvex $\mathcal{H}_\infty$ cost. Computational details are given in \ref{['appendix:details-of-figure-1']}.
  • Figure 2: Local geometry of a smooth function $\phi(x)$: (a) a non-stationary point with non-zero gradient $\nabla \phi(x) \neq 0$; (b) (strict) saddle; (c) local maximizer; (d) local minimizer. A general nonconvex function $\phi(x)$ may have all these stationary points. If $\phi(x)$ has hidden convexity and is equipped with an $\mathtt{ECL}$, all non-degenerate stationary points are global minimizers (see \ref{['theorem:ECL-guarantee']}); no saddle and local maximizers exist.
  • Figure 3: A schematic illustration of $\mathtt{ECL}$. Left figure: We begin with the epigraph $\operatorname{epi}_\geq(J)$ of a potentially nonconvex and nonsmooth function $J({\mathsf{K}})$; Middle figure: We lift the epigraph $\operatorname{epi}_\geq(J)$ to a set $\mathcal{L}_{\mathrm{lft}}$ of a higher dimension; Right figure: We identify a smooth and invertible mapping $\Phi$ that maps $\mathcal{L}_{\mathrm{lft}}$ to a "partially" convex set $\mathcal{F}_{\mathrm{cvx}} \times \mathcal{G}_{\mathrm{aux}}$. In many control problems, $\mathcal{F}_{\mathrm{cvx}}$ is a convex set represented by LMIs, while $\mathcal{G}_{\mathrm{aux}}$ can be nonconvex and provides the flexibility of capturing similarity transformations (cf. \ref{['section:applications']}). The schematic here serves an illustrative role, and the full $\mathtt{ECL}$ details are elaborated in \ref{['section:ECL']}.
  • Figure 4: Illustration of nonconvex functions and their transformation in \ref{['example:academic']}.
  • Figure 5: Optimization landscape of the LQR instance in \ref{['example:LQR_ex']}: (a) the original LQR cost \ref{['eq:LQR-ex-1']}; (b) the cost function after convexification \ref{['eq:LQR-ex-1-y']}.
  • ...and 4 more figures

Theorems & Definitions (60)

  • Example 2.1
  • Example 2.2: A simple LQR instance
  • Example 2.3: A simple $\mathcal{H}_\infty$ instance
  • Remark 2.1: LMI/semidefinite representable sets
  • Definition 3.1: Extended Convex Lifting
  • Theorem 3.1
  • proof
  • Remark 3.1: Lifting and auxiliary set in $\mathtt{ECL}$
  • Remark 3.2: When the infimum is achieved
  • Definition 3.2: Non-degenerate points
  • ...and 50 more