100 research outputs found
Optimization under uncertainty and risk: Quadratic and copositive approaches
Robust optimization and stochastic optimization are the two main paradigms for dealing with the uncertainty inherent in almost all real-world optimization problems. The core principle of robust optimization is the introduction of parameterized families of constraints. Sometimes, these complicated semi-infinite constraints can be reduced to finitely many convex constraints, so that the resulting optimization problem can be solved using standard procedures. Hence flexibility of robust optimization is limited by certain convexity requirements on various objects. However, a recent strain of literature has sought to expand applicability of robust optimization by lifting variables to a properly chosen matrix space. Doing so allows to handle situations where convexity requirements are not met immediately, but rather intermediately.
In the domain of (possibly nonconvex) quadratic optimization, the principles of copositive optimization act as a bridge leading to recovery of the desired convex structures. Copositive optimization has established itself as a powerful paradigm for tackling a wide range of quadratically constrained quadratic optimization problems, reformulating them into linear convex-conic optimization problems involving only linear constraints and objective, plus constraints forcing membership to some matrix cones, which can be thought of as generalizations of the positive-semidefinite matrix cone. These reformulations enable application of powerful optimization techniques, most notably convex duality, to problems which, in their original form, are highly nonconvex.
In this text we want to offer readers an introduction and tutorial on these principles of copositive optimization, and to provide a review and outlook of the literature that applies these to optimization problems involving uncertainty
Hessian barrier algorithms for linearly constrained optimization problems
In this paper, we propose an interior-point method for linearly constrained
optimization problems (possibly nonconvex). The method - which we call the
Hessian barrier algorithm (HBA) - combines a forward Euler discretization of
Hessian Riemannian gradient flows with an Armijo backtracking step-size policy.
In this way, HBA can be seen as an alternative to mirror descent (MD), and
contains as special cases the affine scaling algorithm, regularized Newton
processes, and several other iterative solution methods. Our main result is
that, modulo a non-degeneracy condition, the algorithm converges to the
problem's set of critical points; hence, in the convex case, the algorithm
converges globally to the problem's minimum set. In the case of linearly
constrained quadratic programs (not necessarily convex), we also show that the
method's convergence rate is for some
that depends only on the choice of kernel function (i.e., not on the problem's
primitives). These theoretical results are validated by numerical experiments
in standard non-convex test functions and large-scale traffic assignment
problems.Comment: 27 pages, 6 figure
Active set complexity of the Away-step Frank-Wolfe Algorithm
In this paper, we study active set identification results for the away-step
Frank-Wolfe algorithm in different settings. We first prove a local
identification property that we apply, in combination with a convergence
hypothesis, to get an active set identification result. We then prove, in the
nonconvex case, a novel convergence rate result and active set
identification for different stepsizes (under suitable assumptions on the set
of stationary points). By exploiting those results, we also give explicit
active set complexity bounds for both strongly convex and nonconvex objectives.
While we initially consider the probability simplex as feasible set, in the
appendix we show how to adapt some of our results to generic polytopes.Comment: 23 page
Hessian barrier algorithms for linearly constrained optimization problems
International audienceIn this paper, we propose an interior-point method for linearly constrained-and possibly nonconvex-optimization problems. The method-which we call the Hessian barrier algorithm (HBA)-combines a forward Euler discretization of Hessian-Riemannian gradient flows with an Armijo backtracking step-size policy. In this way, HBA can be seen as an alternative to mirror descent, and contains as special cases the affine scaling algorithm, regularized Newton processes, and several other iterative solution methods. Our main result is that, modulo a nondegeneracy condition, the algorithm converges to the problem's critical set; hence, in the convex case, the algorithm converges globally to the problem's minimum set. In the case of linearly constrained quadratic programs (not necessarily convex), we also show that the method's convergence rate is for some that depends only on the choice of kernel function (i.e., not on the problem's primi-tives). These theoretical results are validated by numerical experiments in standard nonconvex test functions and large-scale traffic assignment problems
On Tractable Convex Relaxations of Standard Quadratic Optimization Problems under Sparsity Constraints
Standard quadratic optimization problems (StQPs) provide a versatile
modelling tool in various applications. In this paper, we consider StQPs with a
hard sparsity constraint, referred to as sparse StQPs. We focus on various
tractable convex relaxations of sparse StQPs arising from a mixed-binary
quadratic formulation, namely, the linear optimization relaxation given by the
reformulation-linearization technique, the Shor relaxation, and the relaxation
resulting from their combination. We establish several structural properties of
these relaxations in relation to the corresponding relaxations of StQPs without
any sparsity constraints, and pay particular attention to the rank-one feasible
solutions retained by these relaxations. We then utilize these relations to
establish several results about the quality of the lower bounds arising from
different relaxations. We also present several conditions that ensure the
exactness of each relaxation.Comment: Technical Report, School of Mathematics, The University of Edinburgh,
Edinburgh, EH9 3FD, Scotland, United Kingdo
First-order Methods for the Impatient: Support Identification in Finite Time with Convergent Frank--Wolfe Variants
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