On nonlinear optimization since 1959

This view of the development of algorithms for nonlinear optimization is based on the research that has been of particular interest to the author since 1959, including several of his own contributions. After a brief survey of classical methods, which may require good starting points in order to converge successfully, the huge impact of variable … Read more

A Note on Complexity of Traveling Tournament Problem

Sports league scheduling problems have gained considerable amount of attention in recent years due to its huge applications and challenges. Traveling Tournament problem, proposed by Trick et. al. (2001), is a problem of scheduling round robin leagues which minimizes the total travel distance maintaining some constraints on consecutive home and away matches. No good algorithm … Read more

Asymptotic expansion for the solution of a penalized control constrained semilinear elliptic problems

In this work we consider the optimal control problem of a semilinear elliptic PDE with a Dirichlet boundary condition, where the control variable is distributed over the domain and is constrained to be nonnegative. The approach is to consider an associated parametrized family of penalized problems, whose solutions define a central path converging to the … Read more

The Mcf-Separator – Detecting and Exploiting Multi-Commodity Flow Structures in MIPs

Given a general mixed integer program (MIP), we automatically detect block structures in the constraint matrix together with the coupling by capacity constraints arising from multi-commodity flow formulations. We identify the underlying graph and generate cutting planes based on cuts in the detected network. Our implementation adds a separator to the branch-and-cut libraries of Scip … Read more

A sufficiently exact inexact Newton step based on reusing matrix information

Newton’s method is a classical method for solving a nonlinear equation $F(z)=0$. We derive inexact Newton steps that lead to an inexact Newton method, applicable near a solution. The method is based on solving for a particular $F'(z_{k’})$ during $p$ consecutive iterations $k=k’,k’+1,\dots,k’+p-1$. One such $p$-cycle requires $2^p-1$ solves with the matrix $F'(z_{k’})$. If matrix … Read more

Concrete Structure Design Using Mixed-Integer Nonlinear Programming with Complementarity Constraints

We present a mixed-integer nonlinear programming (MINLP) formulation to achieve minimum-cost designs for reinforced concrete (RC) structures that satisfy building code requirements. The objective function includes material and labor costs for concrete, steel reinforcing bars, and formwork according to typical contractor methods. Restrictions enforce correct geometry of the cross-section dimensions for each element and relative … Read more

A Multi-Product Risk-Averse Newsvendor with Law Invariant Coherent Measures of Risk

We consider a multi-product newsvendor under the law-invariant coherent risk measures. We first establish a few fundamental properties of the model regarding the convexity of the problem, the symmetry of the solution and the impact of risk aversion. Specifically, we show that for identical products with independent demands, increased risk aversion leads to decreased orders. … Read more

A Collection of 1,300 Dynamical Systems for Testing Data Fitting, Optimal Control, Experimental Design, Identification, Simulation or Similar Software – User’s Guide

We describe a collection of test problems which have been used to develop and test data fitting software for identifying parameters in explicit model functions, dynamical systems of equations, Laplace transformations, systems of ordinary differential equations, differential algebraic equations, or systems of one-dimensional time-dependent partial differential equations with or without algebraic equations. The test cases … Read more

Identifying Active Manifolds in Regularization Problems

In this work we consider the problem $\min_x \{ f(x) + P(x) \}$, where $f$ is $\mathcal{C}^2$ and $P$ is nonsmooth, but contains an underlying smooth substructure. Specifically, we assume the function $P$ is prox-regular partly smooth with respect to a active manifold $\M$. Recent work by Tseng and Yun \cite{tseng-yun-2009}, showed that such a … Read more

MathOptimizer: A nonlinear optimization package for Mathematica users

Mathematica is an advanced software system that enables symbolic computing, numerics, program code development, model visualization and professional documentation in a unified framework. Our MathOptimizer software package serves to solve global and local optimization models developed using Mathematica. We introduce MathOptimizer’s key features and discuss its usage options that support a range of operational modes. … Read more