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Projection onto hyperbolicity cones and beyond: a dual Frank-Wolfe approach

Takayuki Nagano, Bruno F. Lourenço, Akiko Takeda

Abstract

We discuss the problem of projecting a point onto an arbitrary hyperbolicity cone from both theoretical and numerical perspectives. While hyperbolicity cones are furnished with a generalization of the notion of eigenvalues, obtaining closed form expressions for the projection operator as in the case of semidefinite matrices is an elusive endeavour. To address that we propose a Frank-Wolfe method to handle this task and, more generally, strongly convex optimization over closed convex cones. One of our innovations is that the Frank-Wolfe method is actually applied to the dual problem and, by doing so, subproblems can be solved in closed-form using minimum eigenvalue functions and conjugate vectors. To test the validity of our proposed approach, we present numerical experiments where we check the performance of alternative approaches including interior point methods and an earlier accelerated gradient method proposed by Renegar. We also show numerical examples where the hyperbolic polynomial has millions of monomials. Finally, we also discuss the problem of projecting onto p-cones which, although not hyperbolicity cones in general, are still amenable to our techniques.

Projection onto hyperbolicity cones and beyond: a dual Frank-Wolfe approach

Abstract

We discuss the problem of projecting a point onto an arbitrary hyperbolicity cone from both theoretical and numerical perspectives. While hyperbolicity cones are furnished with a generalization of the notion of eigenvalues, obtaining closed form expressions for the projection operator as in the case of semidefinite matrices is an elusive endeavour. To address that we propose a Frank-Wolfe method to handle this task and, more generally, strongly convex optimization over closed convex cones. One of our innovations is that the Frank-Wolfe method is actually applied to the dual problem and, by doing so, subproblems can be solved in closed-form using minimum eigenvalue functions and conjugate vectors. To test the validity of our proposed approach, we present numerical experiments where we check the performance of alternative approaches including interior point methods and an earlier accelerated gradient method proposed by Renegar. We also show numerical examples where the hyperbolic polynomial has millions of monomials. Finally, we also discuss the problem of projecting onto p-cones which, although not hyperbolicity cones in general, are still amenable to our techniques.
Paper Structure (36 sections, 22 theorems, 130 equations, 2 figures, 11 tables, 2 algorithms)

This paper contains 36 sections, 22 theorems, 130 equations, 2 figures, 11 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related works
  3. On the generalized Lax conjecture
  4. Outline of this work
  5. Preliminaries
  6. Hyperbolic polynomials
  7. Derivative relaxations
  8. Generalized minimum eigenvalue functions
  9. The Frank-Wolfe Method
  10. Distance function and projection under the norm induced by a hyperbolic polynomial
  11. The distance function to a hyperbolicity cone
  12. Partial results on the projection operator
  13. Limitations and further discussion
  14. A FW algorithm for strongly convex optimization over regular cones
  15. Optimality conditions for \ref{['prob:primal']} and \ref{['prob:dual']}
  16. ...and 21 more sections

Key Result

Theorem 2.1

renegar2004hyperbolic Let $\Lambda$ be a hyperbolicity cone. Define $\partial^r \Lambda = \{ x \in \Lambda \mid \mathrm{mult}(x) = r \}$. Then, for $r \geq 2$, Also,

Figures (2)

  • Figure 1: Frank-Wolfe gap and relative function values log-log plots for the cases $(n,k) \in \{(20,10),(30,15)\}$, $10$ instances each. For relative function values (the plots on the right), since $\hat{f}_{\text{opt}}$ is the best solution obtained during the $10$ seconds, it is natural that the relative error computed empirically goes to $0$. Still, the fact that the graph is almost a straight line before that suggests that the convergence is indeed sublinear as predicted by \ref{['eq:obj_vals_convergence']}.
  • Figure : The Frank-Wolfe method frank1956algorithm

Theorems & Definitions (47)

  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • Definition 3.1: Isometric hyperbolic polynomial, BGLS01
  • Example 3.2: The hyperbolic norm $\left\|\cdot\right\|_p$, the nonnegative orthant, the semidefinite cone and symmetric cones
  • Lemma 3.3
  • proof
  • ...and 37 more