A primal-dual active set method for quadratic problems with bound constraints is presented which extends the infeasible active set approach of Kunisch and Rendl [17]. Based on a guess of the active set, a primal-dual pair (x,α) is computed that satisfies stationarity and the complementary condition. If x is not feasible, the variables connected to the infeasibilities are added to the active set...

In this paper, we present a simpler proof of the result of Tsuchiya and Muramatsu on the convergence of the primal affine scaling method. We show that the primal sequence generated by the method converges to the interior of the optimum face and the dual sequence to the analytic center of the optimal dual face, when the step size implemented in the procedure is bounded by 2/3. We also prove the ...

In this lecture, the focus is on primal dual method and its application in designing exact algorithms and approximation algorithms for combinatorial optimization problems. It is a general framework which can solve many problems rather systematically. We first introduce the principles of primal dual program, and then show one example, weighted bipartite matching for designing exact algorithm and...

We present an optimization approach for linear SVMs based on a stochasticprimal-dual approach, where the primal step is akin to an importance-weightedSGD, and the dual step is a stochastic update on the importance weights. Thisyields an optimization method with a sublinear dependence on the training setsize, and the first method for learning linear SVMs with runtime less the...

The inexact primal-dual interior point method which is discussed in this paper chooses a new iterate along an approximation to the Newton direction. The method is the Kojima, Megiddo, and Mizuno globally convergent infeasible interior point algorithm The inexact variation is shown to have the same convergence properties accepting a residual in both the primal and dual Newton step equation also ...

Although the Lagrangian method is a powerful dual search approach in integer programming, it often fails to identify an optimal solution of the primal problem. The p-th power Lagrangian method developed in this paper ooers a success guarantee for the dual search in generating an optimal solution of the primal integer programming problem in an equivalent setting via two key transformations. One ...

We consider a primal-dual short-step interior-point method for conic convex optimization problems for which exact evaluation of the gradient and Hessian of the primal and dual barrier functions is either impossible or prohibitively expensive. As our main contribution, we show that if approximate gradients and Hessians of the primal barrier function can be computed, and the relative errors in su...

We show, using elementary considerations, that a modified barrier function method for the solution of convex programming problems converges for any fixed positive setting of the barrier parameter. With mild conditions on the primal and dual feasible regions, we show how to use the modified barrier function method to obtain primal and dual optimal solutions, even in the presence of degeneracy. W...

A numerical method is proposed for optimal robust control synthesis. The method applies to the case when the coefficients of the characteristic polynomial depend linearly on the uncertain parameters. A primal/dual pair of infinite-dimensional convex problems is solved by successive finite-dimensional approximations. The primal/dual pair has no duality gap, and both upper and lower bounds produc...