In this paper, a primal-dual interior point method is proposed for general constrained optimization, which incorporated a penalty function and a kind of new identification technique of the active set. At each iteration, the proposed algorithm only needs to solve two or three reduced systems of linear equations with the same coefficient matrix. The size of systems of linear equations can be decreased due to the introduction of the working set, which is an estimate of the active set. The penalty parameter is automatically updated and the uniformly positive definiteness condition on the Hessian approximation of the Lagrangian is relaxed. The proposed algorithm possesses global and superlinear convergence under some mild conditions. Finally, some preliminary numerical results are reported.

We propose an efficient gradient-type algorithm to solve the fourth-order LLT denoising model for both gray-scale and vector-valued images. Based on the primal-dual formulation of the original nondifferentiable model, the new algorithm updates the primal and dual variables alternately using the gradient descent/ascent flows. Numerical examples are provided to demonstrate the superiority of our algorithm.

We consider a special packing-covering pair of problems. Thepacking problem is a natural generalization of finding a(weighted) maximum independent set in an interval graph, thecovering problem generalizes the problem of finding a (weighted)minimum clique cover in an interval graph. The problem pairinvolves weights and capacities; we consider the case of unitweights and the case of unit capacities. In each case we describea simple algorithm that outputs a solution to the packing problemand to the covering problem that are within a factor of 2 of eachother. Each of these results implies an approximative min-maxresult. For the general case of arbitrary weights and capacitieswe describe an LP-based (2 + ε)-approximation algorithm forthe covering problem. Finally, we show that, unlessP = NP, the covering problem cannot be approximated inpolynomial time within arbitrarily good precision.