/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Convex Analysis: (pms-28)

Many optimal decision problems in scientific, engineering, and public sector applications involve both discrete decisions and nonlinear system dynamics that affect the quality of the final design or plan. These decision problems lead to mixed-integer nonlinear programming (MINLP) problems that combine the combinatorial difficulty of optimizing over discrete variable sets with the challenges of handling nonlinear functions. We review models and applications of MINLP, and survey the state of the art in methods for solving this challenging class of problems.Most solution methods for MINLP apply some form of tree search. We distinguish two broad classes of methods: single-tree and multitree methods. We discuss these two classes of methods first in the case where the underlying problem functions are convex. Classical single-tree methods include nonlinear branch-and-bound and branch-and-cut methods, while classical multitree methods include outer approximation and Benders decomposition. The most efficient class of methods for convex MINLP are hybrid methods that combine the strengths of both classes of classical techniques.Non-convex MINLPs pose additional challenges, because they contain non-convex functions in the objective function or the constraints; hence even when the integer variables are relaxed to be continuous, the feasible region is generally non-convex, resulting in many local minima. We discuss a range of approaches for tackling this challenging class of problems, including piecewise linear approximations, generic strategies for obtaining convex relaxations for non-convex functions, spatial branch-and-bound methods, and a small sample of techniques that exploit particular types of non-convex structures to obtain improved convex relaxations.We finish our survey with a brief discussion of three important aspects of MINLP. First, we review heuristic techniques that can obtain good feasible solution in situations where the search-tree has grown too large or we require real-time solutions. Second, we describe an emerging area of mixed-integer optimal control that adds systems of ordinary differential equations to MINLP. Third, we survey the state of the art in software for MINLP.

/pdf/mixed-integer-nonlinear-optimization-3yf57zxyv3.pdf

Mixed-integer nonlinear optimization

https://link.springer.com/content/pdf/10.1057%2Fjors.1996.159.pdf

Modeling and Analysis of Stochastic Systems

We consider the numerical calculation of several matrix eigenvalue problems which require some manipulation before the standard algorithms may be used. This includes finding the stationary values of a quadratic form subject to linear constraints and determining the eigenvalues of a matrix which is modified by a matrix of rank one. We also consider several inverse eigenvalue problems. This includes the problem of determining the coefficients for the Gauss–Radau and Gauss–Lobatto quadrature rules. In addition, we study several eigenvalue problems which arise in least squares.

Some modified matrix eigenvalue problems.

Autonomous vessels: State of the art and potential opportunities in logistics

We study the convex hull of the intersection of a convex set E and a disjunctive set. This intersection is at the core of solution techniques for Mixed Integer Convex Optimization. We prove that if there exists a cone K (resp., a cylinder C) that has the same intersection with the boundary of the disjunction as E, then the convex hull is the intersection of E with K (resp., C).The existence of such a cone (resp., a cylinder) is difficult to prove for general conic optimization. We prove existence and unicity of a second order cone (resp., a cylinder), when E is the intersection of an affine space and a second order cone (resp., a cylinder). We also provide a method for finding that cone, and hence the convex hull, for the continuous relaxation of the feasible set of a Mixed Integer Second Order Cone Optimization (MISOCO) problem, assumed to be the intersection of an ellipsoid with a general linear disjunction. This cone provides a new conic cut for MISOCO that can be used in branch-and-cut algorithms for MISOCO problems.

/pdf/a-conic-representation-of-the-convex-hull-of-disjunctive-173pz3z1kg.pdf

A Conic Representation of the Convex Hull of Disjunctive Sets and Conic Cuts for Integer Second Order Cone Optimization

http://www.optimization-online.org/DB_FILE/2012/08/3563.pdf

On families of quadratic surfaces having fixed intersections with two hyperplanes

Mixed Integer Second Order Cone Optimization, Disjunctive Conic Cuts: Theory and experiments

This paper describes jMarkov, an object-oriented framework designed to facilitate the construction and analysis of large-scale Markov Chains. The object-oriented design allows a natural translation from a conceptual mathematical model to a computer representation. Additionally, the framework provides the user with the solvers to analyze both the steady-state and the transient behaviors, but its design is flexible enough so new solvers can be implemented. Finally, jMarkov provides the tools to model Quasi-Birth and Death Processes with the same philosophy used to construct general Markov Chains.

Julio C. Góez

Papers

Autonomous vessels: State of the art and potential opportunities in logistics

A Conic Representation of the Convex Hull of Disjunctive Sets and Conic Cuts for Integer Second Order Cone Optimization

On families of quadratic surfaces having fixed intersections with two hyperplanes

Mixed Integer Second Order Cone Optimization, Disjunctive Conic Cuts: Theory and experiments

jMarkov: an object-oriented framework for modeling and analyzing Markov chains and QBDs