Parametric Computation of Minimum-Cost Flows with Piecewise Quadratic Costs

We develop algorithms solving parametric flow problems with separable, continuous, piecewise quadratic, and strictly convex cost functions. The parameter to be considered is a common multiplier on the demand of all nodes. Our algorithms compute a family of flows that are each feasible for the respective demand and minimize the costs among the feasible flows for that demand. For single commodity networks with homogenous cost functions, our algorithm requires one matrix multiplication for the initialization, a rank 1 update for each nondegenerate step and the solution of a convex quadratic program for each degenerate step. For nonhomogeneous cost functions, the initialization requires the solution of a convex quadratic program instead. For multi-commodity networks, both the initialization and every step of the algorithm require the solution of a convex program. As each step is mirrored by a breakpoint in the output this yields output-polynomial algorithms in every case.

The Traveling Salesman Problem

The chapter presents a traveling salesman problem, its network properties, convex quadratic formulation, and the solution. In this chapter, it is shown that adding or subtracting a constant to all arcs with special features in a traveling salesman problem (TSP) network model does not change an optimal solution of the TSP. It is also shown that adding or subtracting a constant to all arcs emanating from the same node in a TSP network does not change the TSP optimal solution. In addition, a minimal spanning tree is used to detect sub-tours, and then sub-tour elimination constraints are generated. A convex quadratic program is constructed from the formulated linear integer model of the TSP network. Interior point algorithms are then applied to solve the TSP in polynomial time.

Self-determination motivation theory in R: The software package SDT

This paper presents the technical details of the software package SDT in the R computing and graphics environment, implementing a convex quadratic program that was recently proposed in the literature on self-determination theory of human motivation. Three main features are addressed, with their accompanying code for computation in R: first, the application of the quadratic program and corresponding code for the analysis of the extent of motivation internalization or externalization; second, for exploring the simplex structure assumption of motivation; and third, for adjusting the confounded scoring protocol, called the self-determination or relative autonomy index, to account for the mixture of internal motivation and external motivation. We describe the functions of the R package SDT. The computations are demonstrated with example data accompanying the package, so researchers can run the methodology on their own datasets.

Self-determination motivation theory in R: The software package S D T

This paper presents the technical details of the software package SDT in the R computing and graphics environment, implementing a convex quadratic program that was recently proposed in the literature on self-determination theory of human motivation. Three main features are addressed, with their accompanying code for computation in R: first, the application of the quadratic program and corresponding code for the analysis of the extent of motivation internalization or externalization; second, for exploring the simplex structure assumption of motivation; and third, for adjusting the confounded scoring protocol, called the self-determination or relative autonomy index, to account for the mixture of internal motivation and external motivation. We describe the functions of the R package SDT. The computations are demonstrated with example data accompanying the package, so researchers can run the methodology on their own datasets.

Solving nonmonotone affine variational inequalities problem by DC programming and DCA

In this paper, we consider an optimization model for solving the nonmonotone affine variational inequalities problem (AVI). It is formulated as a DC (Difference of Convex functions) program for which DCA (DC Algorithms) are applied. The resulting DCA are simple: it consists of solving successive convex quadratic program. Numerical experiments on several test problems illustrate the efficiency of the proposed approach in terms of the quality of the obtained solutions and the speed of convergence.

On convex relaxation of graph isomorphism

We consider the problem of exact and inexact matching of weighted undirected graphs, in which a bijective correspondence is sought to minimize a quadratic weight disagreement. This computationally challenging problem is often relaxed as a convex quadratic program, in which the space of permutations is replaced by the space of doubly stochastic matrices. However, the applicability of such a relaxation is poorly understood. We define a broad class of friendly graphs characterized by an easily verifiable spectral property. We prove that for friendly graphs, the convex relaxation is guaranteed to find the exact isomorphism or certify its inexistence. This result is further extended to approximately isomorphic graphs, for which we develop an explicit bound on the amount of weight disagreement under which the relaxation is guaranteed to find the globally optimal approximate isomorphism. We also show that in many cases, the graph matching problem can be further harmlessly relaxed to a convex quadratic program with only n separable linear equality constraints, which is substantially more efficient than the standard relaxation involving 2n equality and n2 inequality constraints. Finally, we show that our results are still valid for unfriendly graphs if additional information in the form of seeds or attributes is allowed, with the latter satisfying an easy to verify spectral characteristic.