A conic formulation for hBcnorm optimization

In this paper, we formulate the $l_p$-norm optimization problem as a conic optimization problem, derive its standard duality properties and show it can be solved in polynomial time. We first define an ad hoc closed convex cone, study its properties and derive its dual. This allows us to express the standard $l_p$-norm optimization primal problem … Read more

Proving strong duality for geometric optimization using a conic formulation

Geometric optimization is an important class of problems that has many applications, especially in engineering design. In this article, we provide new simplified proofs for the well-known associated duality theory, using conic optimization. After introducing suitable convex cones and studying their properties, we model geometric optimization problems with a conic formulation, which allows us to … Read more

A practical general approximation criterion for methods of multipliers based on Bregman distances

This paper demonstrates that for generalized methods of multipliers for convex programming based on Bregman distance kernels — including the classical quadratic method of multipliers — the minimization of the augmented Lagrangian can be truncated using a simple, generally implementable stopping criterion based only on the norms of the primal iterate and the gradient (or … Read more

A BFGS-IP algorithm for solving strongly convex optimization problems with feasibility enforced by an exact penalty approach

This paper introduces and analyses a new algorithm for minimizing a convex function subject to a finite number of convex inequality constraints. It is assumed that the Lagrangian of the problem is strongly convex. The algorithm combines interior point methods for dealing with the inequality constraints and quasi-Newton techniques for accelerating the convergence. Feasibility of … Read more

Newton Algorithms for Large-Scale Strictly Convex Separable Network Optimization

In this work we summarize the basic elements of primal and dual Newton algorithms for network optimization with continuously differentiable (strictly) convex arc cost functions. Both the basic mathematics and implementation are discussed, and hints to important tuning details are made. The exposition assumes that the reader posseses a significant level of prior knowledge in … Read more

Two properties of condition numbers for convex programs via implicitly defined barrier functions

We study two issues on condition numbers for convex programs: one has to do with the growth of the condition numbers of the linear equations arising in interior-point algorithms; the other deals with solving conic systems and estimating their distance to infeasibility. These two issues share a common ground: the key tool for their development … Read more

An infeasible active set method for convex problems with simple bounds

A primal-dual active set method for convex quadratic problems with bound constraints is presented. Based on a guess on the active set, a primal-dual pair $(x,s)$ is computed that satisfies the first order optimality condition and the complementarity condition. If $(x,s)$ is not feasible, a new active set is determined, and the process is iterated. … Read more