Polyhedral studies on the convex recoloring problem

A coloring of the vertices of a graph G is convex if, for each assigned color d, the vertices with color d induce a connected subgraph of G. We address the convex recoloring problem, defined as follows. Given a graph G and a coloring of its vertices, recolor a minimum number of vertices of G, so that the resulting coloring is convex. This problem is known to be NP-hard even when G is a path. We show an integer programming formulation for the weighted version of this problem on arbitrary graphs, and then specialize it for trees. We study the facial structure of the polytope defined as the convex hull of the integer points satisfying the restrictions of the proposed ILP formulation, present several classes of facet-defining inequalities and discuss separation algorithms.     

Initialization in semidefinite programming via a self-dual skew-symmetric embedding

The formulation of interior point algorithms for semidefinite programming has become an active research area, following the success of the methods for large-scale linear programming. Many interior point methods for linear programming have now been extended to the more general semidefinite case, but the initialization problem remained unsolved.In this paper we show that the initialization strategy of embedding the problem in a self-dual skew-symmetric problem can also be extended to the semidefinite case. This method also provides a solution for the initialization of quadratic programs and it is applicable to more general convex problems with conic formulation.     

Fuzzy portfolio optimization a quadratic programming approach

The topic of this paper is as, the title shows, to introduce the formulation of fuzzy portfolio optimization problem as a convex quadratic programming approach and then give an acceptable solution to such problem. A numerical example included in the support of this paper for illustration.     

On Linear Fractional Programming Problem and its Computation Using a Neural Network Model

In this paper we consider linear fractional programming problem and look at its linear complementarity formulation. In the literature, uniqueness of solution of a linear fractional programming problem is characterized through strong quasiconvexity. We present another characterization of uniqueness through complementarity approach and show that the solution set of a fractional programming problem is convex. Finally we formulate the complementarity condition as a set of dynamical equations and prove certain results involving the neural network model. A computational experience is also reported.      

An Exact Reformulation Algorithm for Large Nonconvex NLPs Involving Bilinear Terms

Many nonconvex nonlinear programming (NLP) problems of practical interest involve bilinear terms and linear constraints, as well as, potentially, other convex and nonconvex terms and constraints. In such cases, it may be possible to augment the formulation with additional linear constraints (a subset of Reformulation-Linearization Technique constraints) which do not affect the feasible region of the original NLP but tighten that of its convex relaxation to the extent that some bilinear terms may be dropped from the problem formulation. We present an efficient graph-theoretical algorithm for effecting such exact reformulations of large, sparse NLPs. The global solution of the reformulated problem using spatial Branch-and Bound algorithms is usually significantly faster than that of the original NLP. We illustrate this point by applying our algorithm to a set of pooling and blending global optimization problems.     

Maximizing a submodular function by integer programming: Polyhedral results for the quadratic case

We give a new mixed integer programming (MIP) formulation for the quadratic cost partition problem that is derived from a MIP formulation for maximizing a submodular function. Several classes of valid inequalities for the convex hull of the feasible solutions are derived using the valid inequalities for the node packing polyhedron. Facet defining conditions and separation algorithms are discussed and computational results are reported.     

Deriving the Properties of Linear Bilevel Programming via a Penalty Function Approach

For the linear bilevel programming problem, we propose an assumption weaker than existing assumptions, while achieving similar results via a penalty function approach. The results include: equivalence between (i) existence of a solution to the problem, (ii) existence of an exact penalty function approach for solving the problem, and (iii) achievement of the optimal value of the equivalent form of the problem at some vertex of a certain polyhedral convex set. We prove that the assumption is both necessary and sufficient for the linear bilevel programming problem to admit an exact penalty function formulation, provided that the equivalent form of the problem has a feasible solution. A method is given for computing the minimal penalty function parameter value. This method can be executed by solving a set of linear programming problems. Lagrangian duality is also presented.     

A New Self-Dual Embedding Method for Convex Programming

In this paper we introduce a conic optimization formulation to solve constrained convex programming, and propose a self-dual embedding model for solving the resulting conic optimization problem. The primal and dual cones in this formulation are characterized by the original constraint functions and their corresponding conjugate functions respectively. Hence they are completely symmetric. This allows for a standard primal-dual path following approach for solving the embedded problem. Moreover, there are two immediate logarithmic barrier functions for the primal and dual cones. We show that these two logarithmic barrier functions are conjugate to each other. The explicit form of the conjugate functions are in fact not required to be known in the algorithm. An advantage of the new approach is that there is no need to assume an initial feasible solution to start with. To guarantee the polynomiality of the path-following procedure, we may apply the self-concordant barrier theory of Nesterov and Nemirovski. For this purpose, as one application, we prove that the barrier functions constructed this way are indeed self-concordant when the original constraint functions are convex and quadratic. We pose as an open question to find general conditions under which the constructed barrier functions are self-concordant.     

Dual Semidefinite Programs Without Duality Gaps for a Class of Convex Minimax Programs

In this paper, we introduce a new dual program, which is representable as a semidefinite linear programming problem, for a primal convex minimax programming problem, and we show that there is no duality gap between the primal and the dual whenever the functions involved are sum-of-squares convex polynomials. Under a suitable constraint qualification, we derive strong duality results for this class of minimax problems. Consequently, we present applications of our results to robust sum-of-squares convex programming problems under data uncertainty and to minimax fractional programming problems with sum-of-squares convex polynomials. We obtain these results by first establishing sum-of-squares polynomial representations of non-negativity of a convex max function over a system of sum-of-squares convex constraints. The new class of sum-of-squares convex polynomials is an important subclass of convex polynomials and it includes convex quadratic functions and separable convex polynomials. The sum-of-squares convexity of polynomials can numerically be checked by solving semidefinite programming problems whereas numerically verifying convexity of polynomials is generally very hard.     

Convex programming and the lexicographic multicriteria problem

Mathematical programming formulation of the convex lexicographic multi-criteria problems typically lacks a constraint qualification. Therefore the classical Kuhn-tucker theory fails to characterize their optimal solutions. Furthermore, numerical methods for solving the lexicographic problems are virtually nonexistent. This paper shows that using a recent theory of convex programming, which is free of a constraint qualification assumption, it is possible both to characterize and to calculate the optimal solutions of the convex lexicographic problem.     

Numerical solution of a linear bilevel problem

The linear bilevel programming problem in the optimistic formulation is studied. It is reduced to an optimization problem with a nonconvex constraint in the form of a d.c. function (that is, the difference of two convex functions). For this problem, local and global search methods are developed. Numerical experiments performed for numerous specially generated problems, including large-scale ones, demonstrate the efficiency of the proposed approach.     

On the impact of symmetry-breaking constraints on spatial Branch-and-Bound for circle packing in a square

We study the problem of packing equal circles in a square from the mathematical programming point of view. We discuss different formulations, we analyze formulation symmetries, we propose some symmetry breaking constraints and show that not only do they tighten the convex relaxation bound, but they also ease the task of local NLP solution algorithms in finding feasible solutions. We solve the problem by means of a standard spatial Branch-and-Bound implementation, and show that our formulation improvements allow the algorithm to find very good solutions at the root node.     

First-order inertial algorithms involving dry friction damping

We present a Branch-and-Cut algorithm for a class of nonlinear chance-constrained mathematical optimization problems with a finite number of scenarios. Unsatisfied scenarios can enter a recovery mode. This class corresponds to problems that can be reformulated as deterministic convex mixed-integer nonlinear programming problems with indicator variables and continuous scenario variables, but the size of the reformulation is large and quickly becomes impractical as the number of scenarios grows. The Branch-and-Cut algorithm is based on an implicit Benders decomposition scheme, where we generate cutting planes as outer approximation cuts from the projection of the feasible region on suitable subspaces. The size of the master problem in our scheme is much smaller than the deterministic reformulation of the chance-constrained problem. We apply the Branch-and-Cut algorithm to the mid-term hydro scheduling problem, for which we propose a chance-constrained formulation. A computational study using data from ten hydroplants in Greece shows that the proposed methodology solves instances faster than applying a general-purpose solver for convex mixed-integer nonlinear programming problems to the deterministic reformulation, and scales much better with the number of scenarios.

     

Duality Theorems for Separable Convex Programming Without Qualifications

In the research of mathematical programming, duality theorems are essential and important elements. Recently, Lagrange duality theorems for separable convex programming have been studied. Tseng proves that there is no duality gap in Lagrange duality for separable convex programming without any qualifications. In other words, although the infimum value of the primal problem equals to the supremum value of the Lagrange dual problem, Lagrange multiplier does not always exist. Jeyakumar and Li prove that Lagrange multiplier always exists without any qualifications for separable sublinear programming. Furthermore, Jeyakumar and Li introduce a necessary and sufficient constraint qualification for Lagrange duality theorem for separable convex programming. However, separable convex constraints do not always satisfy the constraint qualification, that is, Lagrange duality does not always hold for separable convex programming. In this paper, we study duality theorems for separable convex programming without any qualifications. We show that a separable convex inequality system always satisfies the closed cone constraint qualification for quasiconvex programming and investigate a Lagrange-type duality theorem for separable convex programming. In addition, we introduce a duality theorem and a necessary and sufficient optimality condition for a separable convex programming problem, whose constraints do not satisfy the Slater condition.     

An exterior point polynomial-time algorithm for convex quadratic programming

In this paper an exterior point polynomial time algorithm for convex quadratic programming problems is proposed. We convert a convex quadratic program into an unconstrained convex program problem with a self-concordant objective function. We show that, only with duality, the Path-following method is valid. The computational complexity analysis of the algorithm is given.     

Extension of some results for channel capacity using a generalized information measure

A new formulation for the channel capacity problem is derived by using the duality theory of convex programming. The simple nature of this dual representation is suitable for computational purposes. The results are derived in a unified way by formulating the channel capacity problem as a special case of a general class of concave programming problems involving a generalized information measure recently introduced by Burbea and Rao [10].Research supported by National Science Foundation Grant No. ECS-8604354.     

Approximation of convex functions by projections of polyhedra

A method for approximate solution of minimization problems for multivariable convex functions with convex constraints is proposed. The main idea consists in approximation of the objective function and constraints by piecewise linear functions and subsequent reduction of the initial convex programming problem to a problem of linear programming. We present algorithms constructing approximating polygons for some classes of single variable convex functions. The many-dimensional problem is reduced to a one-dimensional one by an inductive procedure. The efficiency of the method is illustrated by numerical examples.     

Partial linear characterizations of the asymmetric travelling salesman polytope

We consider the linear programming formulation of the asymmetric travelling salesman problem. Several new inequalities are stated which yield a sharper characterization in terms of linear inequalities of the travelling salesman polytope, i.e., the convex hull of tours. In fact, some of the new inequalities as well as some of the well-known subtour elimination constraints are indeed facets of the travelling salesman polytope, i.e., belong to the class of inequalities that uniquely characterize the convex hull of tours to an-city problem.     

Recovering risk-neutral probability density functions from options prices using cubic splines and ensuring nonnegativity

We present a new approach to estimate the risk-neutral probability density function (pdf) of the future prices of an underlying asset from the prices of options written on the asset. The estimation is carried out in the space of cubic spline functions, yielding appropriate smoothness. The resulting optimization problem, used to invert the data and determine the corresponding density function, is a convex quadratic or semidefinite programming problem, depending on the formulation. Both of these problems can be efficiently solved by numerical optimization software.