A Bundle Approach for SDPs with Exact Subgraph Constraints

The 'exact subgraph' approach was recently introduced as a hierarchical scheme to get increasingly tight semidefinite programming relaxations of several NP-hard graph optimization problems. Solving these relaxations is a computational challenge because of the potentially large number of violated subgraph constraints. We introduce a computational framework for these relaxations designed to cope with these difficulties. We suggest a partial Lagrangian dual, and exploit the fact that its evaluation decomposes into two independent subproblems. This opens the way to use the bundle method from non-smooth optimization to minimize the dual function. Computational experiments on the Max-Cut, stable set and coloring problem show the efficiency of this approach

Using a Factored Dual in Augmented Lagrangian Methods for Semidefinite Programming

In the context of augmented Lagrangian approaches for solving semidefinite programming problems, we investigate the possibility of eliminating the positive semidefinite constraint on the dual matrix by employing a factorization. Hints on how to deal with the resulting unconstrained maximization of the augmented Lagrangian are given. We further use the approximate maximum of the augmented Lagrangian with the aim of improving the convergence rate of alternating direction augmented Lagrangian frameworks. Numerical results are reported, showing the benefits of the approach.Comment: 7 page

The merger of populations, the incidence of marriages, and aggregate unhappiness

Let a society’s unhappiness be measured by the aggregate of the levels of relative deprivation of its members. When two societies of equal size, F and M, merge, unhappiness in the merged society is shown to be higher than the sum of the levels of unhappiness in the constituent societies when apart; merger alone increases unhappiness. But when societies F and M merge and marriages are formed such that the number of households in the merged society is equal to the number of individuals in one of the constituent societies, unhappiness in the merged society is shown to be lower than the aggregate unhappiness in the two constituent societies when apart. This result obtains regardless of which individuals from one society form households with which individuals from the other, and even when the marriages have not (or not yet) led to income gains to the married couples from increased efficiency, scale economies, and the like. While there are various psychological reasons for people to become happier when they get married as opposed to staying single, the very formation of households reduces social distress even before any other happiness-generating factors kick in.Merger of populations, Integration of societies, Unhappiness, Marriages, Relative deprivation, Institutional and Behavioral Economics, D0, D10, D31, D63,

The merger of populations, the incidence of marriages, and aggregate unhappiness

Let a society's unhappiness be measured by the aggregate of the levels of relative deprivation of its members. When two societies of equal size, F and M, merge, unhappiness in the merged society is shown to be higher than the sum of the levels of unhappiness in the constituent societies when apart; merger alone increases unhappiness. But when societies F and M merge and marriages are formed such that the number of households in the merged society is equal to the number of individuals in one of the constituent societies, unhappiness in the merged society is shown to be lower than the aggregate unhappiness in the two constituent societies when apart. This result obtains regardless of which individuals from one society form households with which individuals from the other, and even when the marriages have not (or not yet) led to income gains to the married couples from increased efficiency, scale economies, and the like. While there are various psychological reasons for people to become happier when they get married as opposed to staying single, the very formation of households reduces social distress even before any other happiness-generating factors kick in. --Merger of populations,Integration of societies,Unhappiness,Marriages,Relative Deprivation

Stable-Set and Coloring bounds based on 0-1 quadratic optimization

We consider semidefinite relaxations of Stable-Set and Coloring, which are based on quadratic 0-1 optimization. Information about the stability number and the chromatic number is hidden in the objective function. This leads to simplified relaxations which depend mostly on the number of vertices of the graph. We also propose tightenings of the relaxations which are based on the maximal cliques of the underlying graph. Computational results on graphs from the literature show the strong potential of this new approach

A Computational Study of Exact Subgraph Based SDP Bounds for Max-Cut, Stable Set and Coloring

The "exact subgraph" approach was recently introduced as a hierarchical scheme to get increasingly tight semidefinite programming relaxations of several NP-hard graph optimization problems. Solving these relaxations is a computational challenge because of the potentially large number of violated subgraph constraints. We introduce a computational framework for these relaxations designed to cope with these difficulties. We suggest a partial Lagrangian dual, and exploit the fact that its evaluation decomposes into several independent subproblems. This opens the way to use the bundle method from non-smooth optimization to minimize the dual function. Finally computational experiments on the Max-Cut, stable set and coloring problem show the excellent quality of the bounds obtained with this approach.Comment: arXiv admin note: substantial text overlap with arXiv:1902.0534

The Merger of Populations, the Incidence of Marriages,and Aggregate Unhappiness

Let a society’s unhappiness be measured by the aggregate of the levels of relative deprivation of its members. When two societies of equal size, F and M, merge, unhappiness in the merged society is shown to be higher than the sum of the levels of unhappiness in the constituent societies when apart; merger alone increases unhappiness. But when societies F and M merge and marriages are formed such that the number of households in the merged society is equal to the number of individuals in one of the constituent societies, unhappiness in the merged society is shown to be lower than the aggregate unhappiness in the two constituent societies when apart. This result obtains regardless of which individuals from one society form households with which individuals from the other, and even when the marriages have not (or not yet) led to income gains to the married couples from increased efficiency, scale economies, and the like. While there are various psychological reasons for people to become happier when they get married as opposed to staying single, the very formation of households reduces social distress even before any other happiness-generating factors kick in