We consider a general class of problems of minimization of convex integral functionals subject to linear constraints. Using Fenchel duality, we prove the equality of the values of the minimization problem and its associated dual problem. This equality is a variational criterion for the existence of solution to a large class of inverse problems entering the class of generalized Fredholm integral...

We describe an infeasible interior point algorithm for convex minimization problems. The method uses quasi-Newton techniques for approximating the second derivatives and providing superlinear convergence. We propose a new feasibility control of the iterates by introducing shift variables and by penalizing them in the barrier problem. We prove global convergence under standard conditions on the ...

Numerical experiences show that bundle methods are very efficient for solving convex non-smooth optimization problems. In this paper we describe briefly the mathematical background of a bundle method and discuss practical aspects for the numerical implementation. Further, we give a detailed documentation of our implementation and report about numerical tests.

Consider a problem of minimizing a separable, strictly convex, monotone and differentiable function on a convex polyhedron generated by a system of m linear inequalities. The problem has a series-parallel structure, with the variables divided serially into n disjoint subsets, whose elements are considered in parallel. This special structure is exploited in two algorithms proposed here for the a...

In this paper, we present primal-dual approaches based on configuration linear programs to design competitive online algorithms for problems with arbitrarily-grown objective. Non-linear, especially convex, objective functions have been extensively studied in recent years in which approaches relies crucially on the convexity property of cost functions. Besides, configuration linear programs have...

This paper studies convergence properties of regularized Newton methods for minimizing a convex function whose Hessian matrix may be singular everywhere. We show that if the objective function is LC2, then the methods possess local quadratic convergence under a local error bound condition without the requirement of isolated nonsingular solutions. By using a backtracking line search, we globaliz...

We develop an intuitive geometric interpretation of the standard support vector machine (SVM) for classification of both linearly separable and inseparable data and provide a rigorous derivation of the concepts behind the geometry. For the separable case finding the maximum margin between the two sets is equivalent to finding the closest points in the smallest convex sets that contain each clas...

In this paper we extend the standard bundle proximal method for finding the minimum of a convex not necessarily differentiable function on the nonnegative orthant. The strategy consists in approximating the objective function by a piecewise linear convex function and using distance–like functions based on second order homogeneous kernels. First we prove the convergence of this new bundle interi...

In this paper we address the problem of load balancing optimization of telecommunication networks based on multiple spanning tree routing. We focus on two objectives – minimization of the maximum link load and minimization of the network utilization imposing a worst case load value – and we propose two sets of mixed integer programming models defining the optimization problems (where one set is...

Computing maximum a posteriori configuration in a first-order Markov Random Field has become a routinely used approach in computer vision. It is equivalent to minimizing an energy function of discrete variables. In this paper we consider a subclass of minimization problems in which unary and pairwise terms of the energy function are convex. Such problems arise in many vision applications includ...