Minimizing Submodular Functions on Diamonds via Generalized Fractional Matroid Matchings

In this paper we show the first polynomial-time algorithm for the problem of minimizing submodular functions on the product of diamonds. This submodular function minimization problem is reduced to the membership problem for an associated polyhedron, which is equivalent to the optimization problem over the polyhedron, based on the ellipsoid method. The latter optimization problem is solved by polynomial number of solutions of subproblems, each being a generalization of the weighted fractional matroid matching problem. We give a combinatorial polynomial-time algorithm for this optimization problem by extending the result by Gijswijt and Pap [D.~Gijswijt and G.~Pap, An algorithm for weighted fractional matroid matching, J.\ Combin.\ Theory, Ser.~B 103 (2013), 509--520]

A class of linearly constrained nonlinear optimization problems with corner point optimal solutions and applications in finance

We identify a class of linearly constrained nonlinear optimization problems with corner point optimal solutions. These include some special polynomial fractional optimization problems with an objective function equal to the product of some power functions of positive linear functionals subtracting the sum of some power functions of positive linear functionals, divided by the sum of some power functions of positive linear functionals. The powers are required to be all positive integers, and the aggregate power of the product is required to be no larger than the lowest power in both of the two sums. The result has applications to some optimization problems under uncertainty, particularly in finance

A New Approach to the Numerical Solution of Fractional Order Optimal Control Problems

In this article, a new numerical method is proposed for solving a class of fractional order optimal control problems. The fractional derivative is considered in the Caputo sense. This approach is based on a combination of the perturbation homotopy and parameterization methods. The control function u(t) is approximated by polynomial functions with unknown coefficients. This method converts the fractional order optimal control problem to an optimization problem. Numerical results are included to demonstrate the validity and applicability of the method

Solving Fractional Polynomial Problems by Polynomial Optimization Theory

This work aims to introduce the framework of polynomial optimization theory to solve fractional polynomial problems (FPPs). Unlike other widely used optimization frameworks, the proposed one applies to a larger class of FPPs, not necessarily defined by concave and convex functions. An iterative algorithm that is provably convergent and enjoys asymptotic optimality properties is proposed. Numerical results are used to validate its accuracy in the non-asymptotic regime when applied to the energy efficiency maximization in multiuser multiple-input multiple-output communication systems.Comment: 5 pages, 2 figures, 1 table, submitted to Signal Processing Letter

Parametric approaches to fractional programs: Analytical and empirical study

Fractional programming is used to model problems where the objective function is a ratio of functions. A parametric modeling approach provides effective technique for obtaining optimal solutions of these fractional programming problems. Although many heuristic algorithms have been proposed and assessed relative to each other, there are limited theoretical studies on the number of steps to obtain the solution. In this dissertation, I focus on the linear fractional combinatorial optimization problem, a special case of fractional programming where all functions in the objective function and constraints are linear and all variables are binary that model certain combinatorial structures. Two parametric algorithms are considered and the efficiency of the algorithms is investigated both theoretically and computationally. I develop the complexity bounds for these algorithms, and show that they can solve the linear fractional combinatorial optimization problem in polynomial time. In the computational study, the algorithms are used to solve fractional knapsack problem, fractional facility location problem, and fractional transportation problem by comparison to other algorithms (e.g., Newton\u27s method). The relative practical performance measured by the number of function calls demonstrates that the proposed algorithms are fast and robust for solving the linear fractional programs with discrete variables

Bounding extreme events in nonlinear dynamics using convex optimization

We study a convex optimization framework for bounding extreme events in nonlinear dynamical systems governed by ordinary or partial differential equations (ODEs or PDEs). This framework bounds from above the largest value of an observable along trajectories that start from a chosen set and evolve over a finite or infinite time interval. The approach needs no explicit trajectories. Instead, it requires constructing suitably constrained auxiliary functions that depend on the state variables and possibly on time. Minimizing bounds over auxiliary functions is a convex problem dual to the non-convex maximization of the observable along trajectories. This duality is strong, meaning that auxiliary functions give arbitrarily sharp bounds, for sufficiently regular ODEs evolving over a finite time on a compact domain. When these conditions fail, strong duality may or may not hold; both situations are illustrated by examples. We also show that near-optimal auxiliary functions can be used to construct spacetime sets that localize trajectories leading to extreme events. Finally, in the case of polynomial ODEs and observables, we describe how polynomial auxiliary functions of fixed degree can be optimized numerically using polynomial optimization. The corresponding bounds become sharp as the polynomial degree is raised if strong duality and mild compactness assumptions hold. Analytical and computational ODE examples illustrate the construction of bounds and the identification of extreme trajectories, along with some limitations. As an analytical PDE example, we bound the maximum fractional enstrophy of solutions to the Burgers equation with fractional diffusion.Comment: Revised according to comments by reviewers. Added references and rearranged introduction, conclusions, and proofs. 38 pages, 7 figures, 4 tables, 4 appendices, 87 reference

Max-sum diversity via convex programming

Diversity maximization is an important concept in information retrieval, computational geometry and operations research. Usually, it is a variant of the following problem: Given a ground set, constraints, and a function $f(\cdot)$ that measures diversity of a subset, the task is to select a feasible subset $S$ such that $f(S)$ is maximized. The \emph{sum-dispersion} function $f(S) = \sum_{x,y \in S} d(x,y)$, which is the sum of the pairwise distances in $S$, is in this context a prominent diversification measure. The corresponding diversity maximization is the \emph{max-sum} or \emph{sum-sum diversification}. Many recent results deal with the design of constant-factor approximation algorithms of diversification problems involving sum-dispersion function under a matroid constraint. In this paper, we present a PTAS for the max-sum diversification problem under a matroid constraint for distances $d(\cdot,\cdot)$ of \emph{negative type}. Distances of negative type are, for example, metric distances stemming from the $\ell_2$ and $\ell_1$ norm, as well as the cosine or spherical, or Jaccard distance which are popular similarity metrics in web and image search