From Optimization to Control: Quasi Policy Iteration

Recent control algorithms for Markov decision processes (MDPs) have been designed using an implicit analogy with well-established optimization algorithms. In this paper, we make this analogy explicit across four problem classes with a unified solution characterization. This novel framework, in turn, allows for a systematic transformation of algorithms from one domain to the other. In … Read more

Global convergence of a BFGS-type algorithm for nonconvex multiobjective optimization problems

We propose a modified BFGS algorithm for multiobjective optimization problems with global convergence, even in the absence of convexity assumptions on the objective functions. Furthermore, we establish the superlinear convergence of the method under usual conditions. Our approach employs Wolfe step sizes and ensures that the Hessian approximations are updated and corrected at each iteration … Read more

Shape-Changing Trust-Region Methods Using Multipoint Symmetric Secant Matrices

In this work, we consider methods for large-scale and nonconvex unconstrained optimization. We propose a new trust-region method whose subproblem is defined using a so-called “shape-changing” norm together with densely-initialized multipoint symmetric secant (MSS) matrices to approximate the Hessian. Shape-changing norms and dense initializations have been successfully used in the context of traditional quasi Newton … Read more

Polynomial worst-case iteration complexity of quasi-Newton primal-dual interior point algorithms for linear programming

Quasi-Newton methods are well known techniques for large-scale numerical optimization. They use an approximation of the Hessian in optimization problems or the Jacobian in system of nonlinear equations. In the Interior Point context, quasi-Newton algorithms compute low-rank updates of the matrix associated with the Newton systems, instead of computing it from scratch at every iteration. … Read more

A minibatch stochastic Quasi-Newton method adapted for nonconvex deep learning problems

In this study, we develop a limited memory nonconvex Quasi-Newton (QN) method, tailored to deep learning (DL) applications. Since the stochastic nature of (sampled) function information in minibatch processing can affect the performance of QN methods, three strategies are utilized to overcome this issue. These involve a novel progressive trust-region radius update (suitable for stochastic … Read more

A quasi-Newton method with Wolfe line searches for multiobjective optimization

We propose a BFGS method with Wolfe line searches for unconstrained multiobjective optimization problems. The algorithm is well defined even for general nonconvex problems. Global convergence and R-linear convergence to a Pareto optimal point are established for strongly convex problems. In the local convergence analysis, if the objective functions are locally strongly convex with Lipschitz … Read more

LSOS: Line-search Second-Order Stochastic optimization methods for nonconvex finite sums

We develop a line-search second-order algorithmic framework for minimizing finite sums. We do not make any convexity assumptions, but require the terms of the sum to be continuously differentiable and have Lipschitz-continuous gradients. The methods fitting into this framework combine line searches and suitably decaying step lengths. A key issue is a two-step sampling at … Read more

A Proximal Quasi-Newton Trust-Region Method for Nonsmooth Regularized Optimization

We develop a trust-region method for minimizing the sum of a smooth term f and a nonsmooth term h, both of which can be nonconvex. Each iteration of our method minimizes apossibly nonconvex model of f+h in a trust region. The model coincides with f+h in value and subdifferential at the center. We establish global … Read more

A structured modified Newton approach for solving systems of nonlinear equations arising in interior-point methods for quadratic programming

The focus in this work is interior-point methods for quadratic optimization problems with linear inequality constraints where the system of nonlinear equations that arise are solved with Newton-like methods. In particular, the concern is the system of linear equations to be solved at each iteration. Newton systems give high quality solutions but there is an … Read more

Projected-Search Methods for Bound-Constrained Optimization

Projected-search methods for bound-constrained minimization are based on performing a line search along a continuous piecewise-linear path obtained by projecting a search direction onto the feasible region. A potential benefit of a projected-search method is that many changes to the active set can be made at the cost of computing a single search direction. As … Read more