A New Trust Region Method with Simple Model for Large-Scale Optimization

In this paper a new trust region method with simple model for solving large-scale unconstrained nonlinear optimization problems is proposed. By using the generalized weak quasi-Newton equations, we derive several schemes to determine the appropriate scalar matrix as the Hessian approximation. Under some reasonable conditions and the framework of the trust-region method, the global convergence … Read more

A general inertial proximal point algorithm for mixed variational inequality problem

In this paper, we first propose a general inertial \emph{proximal point algorithm} (PPA) for the mixed \emph{variational inequality} (VI) problem. Based on our knowledge, without stronger assumptions, convergence rate result is not known in the literature for inertial type PPAs. Under certain conditions, we are able to establish the global convergence and nonasymptotic $O(1/k)$ convergence … Read more

Inertial Proximal ADMM for Linearly Constrained Separable Convex Optimization

The \emph{alternating direction method of multipliers} (ADMM) is a popular and efficient first-order method that has recently found numerous applications, and the proximal ADMM is an important variant of it. The main contributions of this paper are the proposition and the analysis of a class of inertial proximal ADMMs, which unify the basic ideas of … Read more

The solution of Euclidean norm trust region SQP subproblems via second order cone programs, an overview and elementary introduction

It is well known that convex SQP subproblems with a Euclidean norm trust region constraint can be reduced to second order cone programs for which the theory of Euclidean Jordan-algebras leads to efficient interior-point algorithms. Here, a brief and self-contained outline of the principles of such an implementation is given. All identities relevant for the … Read more

Semi-Infinite Relaxations for the Dynamic Knapsack Problem with Stochastic Item Sizes

We consider a version of the knapsack problem in which an item size is random and revealed only when the decision maker attempts to insert it. After every successful insertion the decision maker can choose the next item dynamically based on the remaining capacity and available items, while an unsuccessful insertion terminates the process. We … Read more

SDDP for multistage stochastic programs: Preprocessing via scenario reduction

Even with recent enhancements, computation times for large-scale multistage problems with risk-averse objective functions can be very long. Therefore, preprocessing via scenario reduction could be considered as a way to significantly improve the overall performance. Stage-wise backward reduction of single scenarios applied to a fixed branching structure of the tree is a promising tool for … Read more

How to Reach his Desires: Variational Rationality and the Equilibrium Problem on Hadamard Manifolds

In this paper we present a sufficient condition for the existence of a solution for an \mbox{equilibrium} problem on an Hadamard manifold and under suitable assumptions on the sectional curvature, we \mbox{propose} a framework for the convergence analysis of a proximal point algorithm to solve this equilibrium \mbox{problem}. Finally, we offer an application to the … Read more

Borwein-Preiss Variational Principle Revisited

In this article, we refine and slightly strengthen the metric space version of the Borwein–Preiss variational principle due to Li, Shi, J. Math. Anal. Appl. 246, 308–319 (2000), clarify the assumptions and conclusions of their Theorem 1 as well as Theorem 2.5.2 in Borwein, Zhu, Techniques of Variational Analysis, Springer (2005) and streamline the proofs. … Read more

A polynomially solvable case of the pooling problem

Answering a question of Haugland, we show that the pooling problem with one pool and a bounded number of inputs can be solved in polynomial time by solving a polynomial number of linear programs of polynomial size. We also give an overview of known complexity results and remaining open problems to further characterize the border … Read more

Quantifying Double McCormick

When using the standard McCormick inequalities twice to convexify trilinear monomials, as is often the practice in modeling and software, there is a choice of which variables to group first. For the important case in which the domain is a nonnegative box, we calculate the volume of the resulting relaxation, as a function of the … Read more