Analysis of a Path Following Method for Nonsmooth Convex Programs

Recently Gilbert, Gonzaga and Karas [2001] constructed examples of ill-behaved central paths for convex programs. In this paper we show that under mild conditions the central path has sufficient smoothness to allow construction of a path-following interior point algorithm for non-differentiable convex programs. We show that starting from a point near the center of the … Read more

Lagrangian Smoothing Heuristic for Max-Cut

This paper presents smoothing heuristics for an NP-hard combinatorial problem based on Lagrangian relaxation. We formulate the Lagrangian dual for this nonconvex quadratic problem and propose eigenvalue nonsmooth unconstrained optimization to solve the dual problem with bundle or subgradient methods. Derived heuristics are considered to obtain good primal solutions through pathfollowing methods using a projected … Read more

Solving the knapsack problem via Z-transform

Given vectors $a,c\in Z^n$ and $b\in Z$, we consider the (unbounded) knapsack optimization problem $P:\,\min\{c’x\,\vert\, a’x=b;\,x\in N^n\}$. We compute the minimum value $p^*$ using techniques from complex analysis, namely Cauchy residue technique to integrate a function in $C^2$, the $Z$-transform of an appropriate function related to $P$. The computational complexity depends on $s:=\sum_{a_j} a_j$, not … Read more

On a class of nonsmooth composite functions

We discuss in this paper a class of nonsmooth functions which can be represented, in a neighborhood of a considered point, as a composition of a positively homogeneous convex function and a smooth mapping which maps the considered point into the null vector. We argue that this is a sufficiently rich class of functions and … Read more

Lower bound for the number of iterations in semidefinite hierarchies for the cut polytope

Hierarchies of semidefinite relaxations for $0/1$ polytopes have been constructed by Lasserre (2001a) and by Lov\’asz and Schrijver (1991), permitting to find the cut polytope of a graph on $n$ nodes in $n$ steps. We show that $\left\lceil {n\over 2} \right\rceil$ iterations are needed for finding the cut polytope of the complete graph $K_n$. Citation … Read more

On graphs with stability number equal to the optimal value of a convex quadratic program

Since the Motzkin-Straus result on the clique number of graphs, published in 1965, where they show that the size of the largest clique in a graph can be obtained by solving a quadratic programming problem, several results on the continuous approach to the determination of the clique number of a graph or, equivalently, to the … Read more

On differentiability of symmetric matrix valued functions

With every real valued function, of a real argument, can be associated a matrix function mapping a linear space of symmetric matrices into itself. In this paper we study directional differentiability properties of such matrix functions associated with directionally differentiable real valued functions. In particular, we show that matrix valued functions inherit semismooth properties of … Read more

Two-connected networks with rings of bounded cardinality

We study the problem of designing at minimum cost a two-connected network such that each edge belongs to a cycle using at most K edges. This problem is a particular case of the two-connected networks with bounded meshes problem studied by Fortz, Labbé and Maffioli. In this paper, we compute a lower bound on the … Read more