Interior Point Methods for Sufficient Horizontal LCP in a Wide Neighborhood of the Central Path with Best Known Iteration Complexity
Three interior point methods are proposed for sufficient horizontal linear complementarity problems (HLCP): a large update path following algorithm, a first order corrector-predictor method, and a second order corrector-predictor method. All algorithms ...
Calmness Modulus of Linear Semi-infinite Programs
Our main goal is to compute or estimate the calmness modulus of the argmin mapping of linear semi-infinite optimization problems under canonical perturbations, i.e., perturbations of the objective function together with continuous perturbations of the right-...
Bounds on Eigenvalues of Matrices Arising from Interior-Point Methods
Interior-point methods feature prominently among numerical methods for inequality-constrained optimization problems, and involve the need to solve a sequence of linear systems that typically become increasingly ill-conditioned with the iterations. To solve ...
On Stochastic Subgradient Mirror-Descent Algorithm with Weighted Averaging
This paper considers stochastic subgradient mirror-descent method for solving constrained convex minimization problems. In particular, a stochastic subgradient mirror-descent method with weighted iterate-averaging is investigated and its per-iterate ...
The Length of the Primal-Dual Path in Moreau--Yosida-Based Path-Following Methods for State Constrained Optimal Control
A priori estimates of the length of the primal-dual path resulting from a Moreau--Yosida approximation of the feasible set for state constrained optimal control problems are derived. These bounds depend on the regularity of the state and the dimension of ...
Multistage Stochastic Decomposition: A Bridge between Stochastic Programming and Approximate Dynamic Programming
Multistate stochastic programs pose some of the more challenging optimization problems. Because such models can become rather intractable in general, it is important to design algorithms that can provide approximations which, in the long run, yield solutions ...
Metric Subregularity of Piecewise Linear Multifunctions and Applications to Piecewise Linear Multiobjective Optimization
In this paper, we consider the metric subregularity property for a piecewise linear multifunction (with respect to a piecewise linear constraint) as well as the weak sharp minimum property for a piecewise linear constrained multiobjective optimization ...
A Filter Method with Unified Step Computation for Nonlinear Optimization
We present a filter line search method for solving general nonlinear and nonconvex optimization problems. The method is of the filter variety but uses a robust (always feasible) subproblem based on an exact penalty function to compute a search direction. ...
A Class of Linear Generalized Equations
Solution stability of a class of linear generalized equations in finite dimensional Euclidean spaces is investigated by means of generalized differentiation. Exact formulas for the Fréchet and the Mordukhovich coderivatives of the normal cone mappings of ...
A Dynamical Approach to an Inertial Forward-Backward Algorithm for Convex Minimization
We introduce a new class of forward-backward algorithms for structured convex minimization problems in Hilbert spaces. Our approach relies on the time discretization of a second-order differential system with two potentials and Hessian-driven damping, ...
Facially Exposed Cones Are Not Always Nice
We address the conjecture proposed by Gábor Pataki that every facially exposed cone is nice. We show that the conjecture is true in the three-dimensional case; however, there exists a four-dimensional counterexample of a cone that is facially exposed but is ...
Rate of Convergence Analysis of Decomposition Methods Based on the Proximal Method of Multipliers for Convex Minimization
This paper presents two classes of decomposition algorithms based on the proximal method of multipliers (PMM) introduced in the mid-1970s by Rockafellar for convex minimization. We first show that the PMM framework is at the root of many past and recent ...
A Semismooth Newton Method with Multidimensional Filter Globalization for $l_1$-Optimization
Due to their property of enhancing the sparsity of solutions, $l_1$-regularized optimization problems have developed into a highly dynamic research area with a wide range of applications. We present a class of methods for $l_1$-regularized optimization ...
A Derivative-Free Trust-Region Method for Biobjective Optimization
We consider unconstrained black-box biobjective optimization problems in which analytic forms of the objective functions are not available and function values can be obtained only through computationally expensive simulations. We propose a new algorithm to ...
Proximal Normal Cone Analysis on Smooth Banach Spaces and Applications
Unlike the Fréchet, Penot, Clarke, and other “order one” notions of normal cones, the notion of proximal normal cone describes variational behavior of “order two,” and most of results on proximal normal cone are established in the Hilbert space framework. A ...
On the Identification of the Optimal Partition of Second Order Cone Optimization Problems
This paper discusses the identification of the optimal partition of second order cone optimization (SOCO). By giving some condition numbers which only depend on the SOCO problem itself, we derive some bounds on the magnitude of the blocks of variables along ...
Stationary Point Sets: Convex Quadratic Optimization Is Universal in Nonlinear Optimization
We investigate the local topological structure that stationary point sets in parametric optimization generically may have. Our main result states that up to stratified isomorphism, any such structure is already present in the small subclass of parametric ...
Operator Preconditioning for a Class of Inequality Constrained Optimal Control Problems
We propose and analyze two strategies for preconditioning linear operator equations that arise in PDE constrained optimal control in the framework of conjugate gradient methods. Our particular focus is on control or state constrained problems, where we ...
Quantitative Stability Analysis of Stochastic Generalized Equations
We consider the solution of a system of stochastic generalized equations (SGE) where the underlying functions are mathematical expectation of random set-valued mappings. SGE has many applications such as characterizing optimality conditions of a nonsmooth ...
Analysis of the Convergence Rate for the Cyclic Projection Algorithm Applied to Basic Semialgebraic Convex Sets
In this paper, we study the rate of convergence of the cyclic projection algorithm applied to finitely many basic semialgebraic convex sets. We establish an explicit convergence rate estimate which relies on the maximum degree of the polynomials that ...
Scalable Nonlinear Programming via Exact Differentiable Penalty Functions and Trust-Region Newton Methods
We present an approach for nonlinear programming based on the direct minimization of an exact differentiable penalty function using trust-region Newton techniques. The approach provides desirable features required for scalability: it can detect and exploit ...