Combinatorial problems in finite fields and Sidon sets

We use Sidon sets to present an elementary method to study some combinatorial problems in finite fields, such as sum product estimates, solvability of some equations and the distribution of their solutions. We obtain classic and more recent results avoiding the use of exponential sums, the usual tool to deal with these problems.     

On the complexity of polyhedral separability

It is NP-complete to recognize whether two sets of points in general space can be separated by two hyperplanes. It is NP-complete to recognize whether two sets of points in the plane can be separated withk lines. For every fixedk in any fixed dimension, it takes polynomial time to recognize whether two sets of points can be separated withk hyperplanes.     

An algorithm to solve polyhedral convex set optimization problems

An algorithm which computes a solution of a set optimization problem is provided. The graph of the objective map is assumed to be given by finitely many linear inequalities. A solution is understood to be a set of points in the domain satisfying two conditions: the attainment of the infimum and minimality with respect to a set relation. In the first phase of the algorithm, a linear vector optimization problem, called the vectorial relaxation, is solved. The resulting pre-solution yields the attainment of the infimum but, in general, not minimality. In the second phase of the algorithm, minimality is established by solving certain linear programs in combination with vertex enumeration of some values of the objective map.     

On the degree and separability of nonconvexity and applications to optimization problems

We study qualitative indications for d.c. representations of closed sets in and functions on Hilbert spaces. The first indication is an index of nonconvexity which can be regarded as a measure for the degree of nonconvexity. We show that a closed set is weakly closed if this indication is finite. Using this result we can prove the solvability of nonconvex minimization problems. By duality a minimization problem on a feasible set in which this indication is low, can be reduced to a quasi-concave minimization over a convex set in a low-dimensional space. The second indication is the separability which can be incorporated in solving dual problems. Both the index of nonconvexity and the separability can be characteristics to “good” d.c. representations. For practical computation we present a notion of clouds which enables us to obtain a good d.c. representation for a class of nonconvex sets. Using a generalized Caratheodory’s theorem we present various applications of clouds.     

Perturbed solutions of variational inequality problems over polyhedral sets

This paper is concerned with variational inequality problems defined over polyhedral sets, which provide a generalization of many diverse problems of mathematical programming, complementarity, and mathematical economics. Differentiability properties of locally unique perturbed solutions to such problems are studied. It is shown that, if a simple sufficient condition is satisfied, then the perturbed solution is locally unique, continuous, and directionally differentiable. Furthermore, under an additional regularity assumption, the perturbed solution is also continuously differentiable.     

Note: An algorithm to solve polyhedral convex set optimization problems

We provide a counterexample to the remark in Löhne and Schrage [An algorithm to solve polyhedral convex set optimization problems, Optimization 62 (2013), pp. 131-141] that every solution of a polyhedral convex set optimization problem is a pre-solution. A correct statement is that every solution of a polyhedral convex set optimization problem obtained by the algorithm SetOpt is a pre-solution. We also show that every finite infimizer and hence every solution of a polyhedral convex set optimization problem contains a pre-solution.     

Combinatorial optimization problems in wireless switch design

The purpose of this paper is to illustrate the diversity of combinatorial problems encountered in the design of wireless switching systems. This is done via a representative selection of examples of real problems along with their associated solution methods. It should be emphasized that all the solution methods presented in this paper are successfully operating in the field at the time of writing. To the memory of my beloved father, French Navy Admiral Christian Sirdey, whose life was cut short by cancer on November the 13th, 2006.     

On polyhedral and second-order cone decompositions of semidefinite optimization problems

We study a cutting-plane method for semidefinite optimization problems, and supply a proof of the method’s convergence, under a boundedness assumption. By relating the method’s rate of convergence to an initial outer approximation’s diameter, we argue the method performs well when initialized with a second-order cone approximation, instead of a linear approximation. We invoke the method to provide bound gaps of 0.5–6.5% for sparse PCA problems with 1000s of covariates, and solve nuclear norm problems over 500 × 500 matrices.     

Cone complementarity problems with finite solution sets

We introduce the notion of a complementary cone and a nondegenerate linear transformation and characterize the finiteness of the solution set of a linear complementarity problem over a closed convex cone in a finite dimensional real inner product space. In addition to the above, other geometrical properties of complementary cones have been explored.     

Tractability of convex vector optimization problems in the sense of polyhedral approximations

There are different solution concepts for convex vector optimization problems (CVOPs) and a recent one, which is motivated from a set optimization point of view, consists of finitely many efficient solutions that generate polyhedral inner and outer approximations to the Pareto frontier. A CVOP with compact feasible region is known to be bounded and there exists a solution of this sense to it. However, it is not known if it is possible to generate polyhedral inner and outer approximations to the Pareto frontier of a CVOP if the feasible region is not compact. This study shows that not all CVOPs are tractable in that sense and gives a characterization of tractable problems in terms of the well known weighted sum scalarization problems.     

A smoothing homotopy method for variational inequality problems on polyhedral convex sets

In this paper, based on the Robinson’s normal equation and the smoothing projection operator, a smoothing homotopy method is presented for solving variational inequality problems on polyhedral convex sets. We construct a new smoothing projection operator onto the polyhedral convex set, which is feasible, twice continuously differentiable, uniformly approximate to the projection operator, and satisfies a special approximation property. It is computed by solving nonlinear equations in a neighborhood of the nonsmooth points of the projection operator, and solving linear equations with only finite coefficient matrices for other points, which makes it very efficient. Under the assumption that the variational inequality problem has no solution at infinity, which is a weaker condition than several well-known ones, the existence and global convergence of a smooth homotopy path from almost any starting point in $R^n$ are proven. The global convergence condition of the proposed homotopy method is same with that of the homotopy method based on the equivalent KKT system, but the starting point of the proposed homotopy method is not necessarily an interior point, and the efficiency is more higher. Preliminary test results show that the proposed method is practicable, effective and robust.     

On a constructive approach to optimality conditions for convex SIP problems with polyhedral index sets

In the paper, we consider a problem of convex Semi-Infinite Programming with an infinite index set in the form of a convex polyhedron. In study of this problem, we apply the approach suggested in our recent paper [Kostyukova OI, Tchemisova TV. Sufficient optimality conditions for convex Semi Infinite Programming. Optim. Methods Softw. 2010;25:279–297], and based on the notions of immobile indices and their immobility orders. The main result of the paper consists in explicit optimality conditions that do not use constraint qualifications and have the form of criterion. The comparison of the new optimality conditions with other known results is provided.     

An affine scaling method for optimization problems with polyhedral constraints

Recently an affine scaling, interior point algorithm ASL was developed for box constrained optimization problems with a single linear constraint (Gonzalez-Lima et al., SIAM J. Optim. 21:361–390, 2011). This note extends the algorithm to handle more general polyhedral constraints. With a line search, the resulting algorithm ASP maintains the global and R-linear convergence properties of ASL. In addition, it is shown that the unit step version of the algorithm (without line search) is locally R-linearly convergent at a nondegenerate local minimizer where the second-order sufficient optimality conditions hold. For a quadratic objective function, a sublinear convergence property is obtained without assuming either nondegeneracy or the second-order sufficient optimality conditions.     

Solving electromagnetic eigenvalue problems in polyhedral domains with nodal finite elements

A few years ago, Costabel and Dauge proposed a variational setting, which allows one to solve numerically the time-harmonic Maxwell equations in 3D polyhedral geometries, with the help of a continuous approximation of the electromagnetic field. In order to remove spurious eigenmodes, their method required a parameterization of the variational formulation. In order to avoid this difficulty, we use a mixed variational setting instead of the parameterization, which allows us to handle the divergence-free constraint on the field in a straightforward manner. The numerical analysis of the method is carried out, and numerical examples are provided to show the efficiency of our approach.     

多面体集下多目标优化问题近似解的若干性质

研究了多目标优化问题的近似解. 首先证明了多面体集是 co-radiant集,并证明了一些性质. 随后研究了多面体集下多目标优化问题近似解的特殊性质.     

The stability of the solution sets for set optimization problems via improvement sets

ABSTRACT

The aim of this paper is to investigate the stability of the solution sets for set optimization problems via improvement sets. Firstly, we consider the relations among the solution sets for optimization problem with set optimization criterion. Then, the closeness and the convexity of solution sets are discussed. Furthermore, the upper semi-continuity, Hausdorff upper semi-continuity and lower semi-continuity of solution mappings to parametric set optimization problems via improvement sets are established under some suitable conditions. These results extend and develop some recent works in this field.