In this paper, we derive the explicit finite time local convergence of Nesterov accelerated the Broyden-Fletcher-Goldfarb-Shanno (NA-BFGS) under the assumption that the objective function is strongly convex, its gradient is Lipschitz continuous, and its Hessian is Lipschitz continuous at the optimal point. We have shown that the rate of convergence of the NA-BFGS method is … Read more
\(\) This paper studies hidden convexity properties associated with constrained optimization problems over the set of rotation matrices \(\text{SO}(n)\). Such problems are nonconvex due to the constraint\(X\in\text{SO}(n)\). Nonetheless, we show that certain linear images of \(\text{SO}(n)\) are convex, opening up the possibility for convex optimization algorithms with provable guarantees for these problems. Our main technical … Read more
In this paper, a class of algorithms is developed for bound-constrained optimization. The new scheme uses the gradient-free line search along bent search paths. Unlike traditional algorithms for bound-constrained optimization, our algorithm ensures that the reduced gradient becomes arbitrarily small. It is also proved that all strongly active variables are found and fixed after finitely … Read more
We consider the problem of optimally designing a system for repeated use under uncertainty. We develop a modeling framework that integrates design and operational phases, which are represented by a mixed-integer program and discounted-cost infinite-horizon Markov decision processes, respectively. We seek to simultaneously minimize the design costs and the subsequent expected operational costs. This problem … Read more
We introduce SILS, a subspace inertial line search algorithm, which is a line search method designed for finding zeros of monotone mappings. At each iteration, a new point is generated in a subspace of the previous points, replacing the one with the largest residual norm. This study analyzes the global convergence and complexity bounds for … Read more
Gradient tracking methods have emerged as one of the most popular approaches for solving decentralized optimization problems over networks. In this setting, each node in the network has a portion of the global objective function, and the goal is to collectively optimize this function. At every iteration, gradient tracking methods perform two operations (steps): (1) … Read more
This paper establishes the iteration-complexity of proximal bundle methods for solving hybrid (i.e., a blend of smooth and nonsmooth) weakly convex composite optimization (HWC-CO) problems. This is done in a unified manner by considering a proximal bundle framework (PBF) based on a generic bundle update scheme which includes various well-known bundle update schemes. In contrast … Read more
We study linear programming relaxations of nonconvex quadratic programs given by the reformulation-linearization technique (RLT), referred to as RLT relaxations. We investigate the relations between the polyhedral properties of the feasible regions of a quadratic program and its RLT relaxation. We establish various connections between recession directions, boundedness, and vertices of the two feasible regions. … Read more
A suitable set of test instances, also known as benchmark problems, is a key ingredient to systematically evaluate numerical solution algorithms for a given class of optimization problems. While in recent years several solution algorithms for the class of multiobjective mixed-integer nonlinear optimization problems have been proposed, there is a lack of a well-established set … Read more
We propose in this paper a Polak-Ribière-Polyak conjugate gradient type method for solving bicriteria optimization problems by avoiding scalarization techniques. Two particular advantages in this contribution are to be noted. First, the suggested descent direction common to both criteria may be directly computed by a given formula without solving any intermediate subproblem. Second, the descent … Read more