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The primary focus of this paper is on designing an inexact first-order algorithm for solving constrained nonlinear optimization problems. By controlling the inexactness of the subproblem solution, we can significantly reduce the…
A class of interior point methods using inexact directions is analysed. The linear system arising in interior point methods for linear programming is reformulated such that the solution is less sensitive to perturbations in the right-hand…
In many contemporary optimization problems such as those arising in machine learning, it can be computationally challenging or even infeasible to evaluate an entire function or its derivatives. This motivates the use of stochastic…
This paper presents a convex optimization-based method for finding the globally optimal solutions of a class of mixed-integer non-convex optimal control problems. We consider problems that are non-convex in the input norm, which is a…
In this paper, we study the linear complementarity problems on extended second order cones. We convert a linear complementarity problem on an extended second order cone into a mixed complementarity problem on the non-negative orthant. We…
Fundamentally, every static program analyser searches for a proof through a combination of heuristics providing candidate solutions and a candidate validation technique. Essentially, the heuristic reduces a second-order problem to a…
Motivated by the expressive power of completely positive programming to encode hard optimization problems, many approximation schemes for the completely positive cone have been proposed and successfully used. Most schemes are based on outer…
We define a weak compatibility condition for the Newest Vertex Bisection algorithm on simplex grids of any dimension and show that using this condition the iterative algorithm terminates successfully. Additionally we provide an O(n)…
We consider the minimization of a continuous function over the intersection of a regular cone with an affine set via a new class of adaptive first- and second-order optimization methods, building on the Hessian-barrier techniques introduced…
In this paper we provide a detailed analysis of the iteration complexity of dual first order methods for solving conic convex problems. When it is difficult to project on the primal feasible set described by convex constraints, we use the…
We propose first order algorithms for convex optimization problems where the feasible set is described by a large number of convex inequalities that is to be explored by subgradient projections. The first algorithm is an adaptation of a…
This article investigates the approximation quality achievable for biobjective minimization problems with respect to the Pareto cone by solutions that are (approximately) optimal with respect to larger ordering cones. When simultaneously…
Optimization in machine learning, both theoretical and applied, is presently dominated by first-order gradient methods such as stochastic gradient descent. Second-order optimization methods, that involve second derivatives and/or second…
We develop a decomposition algorithm for distributionally-robust two-stage stochastic mixed-integer convex cone programs, and its important special case of distributionally-robust two-stage stochastic mixed-integer second order cone…
In this workshop, we present a compact but rigorous introduction to second-order optimality conditions for mathematical programs with equilibrium constraints (MPECs). We start from the classical nonlinear programming template, then explain…
The radius of robust feasibility provides a numerical value for the largest possible uncertainty set that guarantees robust feasibility of an uncertain linear conic program. This determines when the robust feasible set is non-empty.…
We study a cutting-plane method for semidefinite optimization problems (SDOs), and supply a proof of the method's convergence, under a boundedness assumption. By relating the method's rate of convergence to an initial outer approximation's…
Minimization methods that search along a curvilinear path composed of a non-ascent nega- tive curvature direction in addition to the direction of steepest descent, dating back to the late 1970s, have been an effective approach to finding a…
In this paper we consider a class of optimization problems with a strongly convex objective function and the feasible set given by an intersection of a simple convex set with a set given by a number of linear equality and inequality…
Successive quadratic approximations, or second-order proximal methods, are useful for minimizing functions that are a sum of a smooth part and a convex, possibly nonsmooth part that promotes regularization. Most analyses of iteration…