This paper presents an extended version of the separation plane algorithms for subgradient-based finite-dimensional nondifferentiable convex blackbox optimization. The extension introduces additional cuts for epigraph of the conjugate of objective function which improve the convergence of the algorithm. The case of affine cuts is considered in more details and it is shown that it requires … Read more
Quantifying the risk carried by an aggregate position $S_d\defn\sum_{i=1}^d X_i$ comprising many risk factors $X_i$ is fundamental to both insurance and financial risk management. Frechet inequalities quantify the worst-case risk carried by the aggregate position given distributional information concerning its composing factors but without assuming independence. This marginal factor modeling of the aggregate position in … Read more
This paper develops what is essentially a simplified version of the block-iterative operator splitting method already proposed by the author and P. Combettes, but with more general initialization conditions. It then describes one way of implementing this algorithm asynchronously under a computing model inspired by modern HPC environments, which consist of interconnected nodes each having … Read more
A subgradient method is presented for solving general convex optimization problems, the main requirement being that a strictly-feasible point is known. A feasible sequence of iterates is generated, which converges to within user-specified error of optimality. Feasibility is maintained with a line-search at each iteration, avoiding the need for orthogonal projections onto the feasible region … Read more
We derive sharp probability bounds on the tails of a product of symmetric non-negative random variables using only information about their first two moments. If the covariance matrix of the random variables is known exactly, these bounds can be computed numerically using semidefinite programming. If only an upper bound on the covariance matrix is available, … Read more
This paper considers the problem of minimizing an expectation function over a closed convex set, coupled with an expectation constraint on either decision variables or problem parameters. We first present a new stochastic approximation (SA) type algorithm, namely the cooperative SA (CSA), to handle problems with the expectation constraint on devision variables. We show that … Read more
This paper studies the iteration-complexity of a new primal-dual algorithm based on Rockafellar’s proximal method of multipliers (PMM) for solving smooth convex programming problems with inequality constraints. In each step, either a step of Rockafellar’s PMM for a second-order model of the problem is computed or a relaxed extragradient step is performed. The resulting algorithm … Read more
We present a new method for minimizing the sum of a differentiable convex function and an $\ell_1$-norm regularizer. The main features of the new method include: $(i)$ an evolving set of indices corresponding to variables that are predicted to be nonzero at a solution (i.e., the support); $(ii)$ a reduced-space subproblem defined in terms of … Read more
The primal-dual hybrid gradient method (PDHG) originates from the Arrow-Hurwicz method, and it has been widely used to solve saddle point problems, particularly in image processing areas. With the introduction of a combination parameter, Chambolle and Pock proposed a generalized PDHG scheme with both theoretical and numerical advantages. It has been analyzed that except for … Read more
We consider a class of augmented Lagrangian methods for solving convex programming problems with inequality constraints. This class involves a family of penalty functions and specific values of parameters $p,q,\tilde y \in R$ and $c>0$. The penalty family includes the classical modified barrier and the exponential function. The associated proximal method for solving the dual … Read more