The exact worst-case convergence rate of the alternating direction method of multipliers

Recently, semidefinite programming performance estimation has been employed as a strong tool for the worst-case performance analysis of first order methods. In this paper, we derive new non-ergodic convergence rates for the alternating direction method of multipliers (ADMM) by using performance estimation. We give some examples which show the exactness of the given bounds. We … Read more

Revisiting semidefinite programming approaches to options pricing: complexity and computational perspectives

In this paper we consider the problem of finding bounds on the prices of options depending on multiple assets without assuming any underlying model on the price dynamics, but only the absence of arbitrage opportunities. We formulate this as a generalized moment problem and utilize the well-known Moment-Sum-of-Squares (SOS) hierarchy of Lasserre to obtain bounds … Read more

Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere

We study the convergence rate of a hierarchy of upper bounds for polynomial minimization prob-lems, proposed by Lasserre [SIAM J. Optim.21(3) (2011), pp.864-885], for the special case when the feasible set is the unit (hyper)sphere. The upper bound at level r of the hierarchy is defined as the minimal expected value of the polynomial over … Read more

A survey of semidefinite programming approaches to the generalized problem of moments and their error analysis

The generalized problem of moments is a conic linear optimization problem over the convex cone of positive Borel measures with given support. It has a large variety of applications, including global optimization of polynomials and rational functions, options pricing in finance, constructing quadrature schemes for numerical integration, and distributionally robust optimization. A usual solution approach, … Read more

Distributionally robust optimization with polynomial densities: theory, models and algorithms

In distributionally robust optimization the probability distribution of the uncertain problem parameters is itself uncertain, and a fictitious adversary, e.g., nature, chooses the worst distribution from within a known ambiguity set. A common shortcoming of most existing distributionally robust optimization models is that their ambiguity sets contain pathological discrete distribution that give nature too much … Read more

Worst-case convergence analysis of gradient and Newton methods through semidefinite programming performance estimation

We provide new tools for worst-case performance analysis of the gradient (or steepest descent) method of Cauchy for smooth strongly convex functions, and Newton’s method for self-concordant functions. The analysis uses semidefinite programming performance estimation, as pioneered by Drori en Teboulle [Mathematical Programming, 145(1-2):451–482, 2014], and extends recent performance estimation results for the method of … Read more

Polynomial Norms

In this paper, we study polynomial norms, i.e. norms that are the dth root of a degree-d homogeneous polynomial f. We first show that a necessary and sufficient condition for f^(1/d) to be a norm is for f to be strictly convex, or equivalently, convex and positive definite. Though not all norms come from dth … Read more

Solving sparse polynomial optimization problems with chordal structure using the sparse, bounded-degree sum-of-squares hierarchy

The sparse bounded degree sum-of-squares (sparse-BSOS) hierarchy of Weisser, Lasserre and Toh [arXiv:1607.01151,2016] constructs a sequence of lower bounds for a sparse polynomial optimization problem. Under some assumptions, it is proven by the authors that the sequence converges to the optimal value. In this paper, we modify the hierarchy to deal with problems containing equality … Read more

Comparison of Lasserre’s measure–based bounds for polynomial optimization to bounds obtained by simulated annealing

We consider the problem of minimizing a continuous function f over a compact set K. We compare the hierarchy of upper bounds proposed by Lasserre in [SIAM J. Optim. 21(3) (2011), pp. 864-885] to bounds that may be obtained from simulated annealing. We show that, when f is a polynomial and K a convex body, … Read more

On the worst-case complexity of the gradient method with exact line search for smooth strongly convex functions

We consider the gradient (or steepest) descent method with exact line search applied to a strongly convex function with Lipschitz continuous gradient. We establish the exact worst-case rate of convergence of this scheme, and show that this worst-case behavior is exhibited by a certain convex quadratic function. We also extend the result to a noisy … Read more