We introduce a notion of self-concordant smoothing for minimizing the sum of two convex functions, one of which is smooth and the other nonsmooth. The key highlight is a natural property of the resulting problem’s structure that yields a variable metric selection method and a step length rule especially suited to proximal quasi-Newton algorithms. Also, … Read more
This paper is devoted to study the convergence rates of a second-order dynamical system and its corresponding discretization associated with a continuously differentiable bilinearly coupled convex-concave saddle point problem. First, we consider the second-order dynamical system with asymptotically vanishing damping term and show the existence and uniqueness of the trajectories as global twice continuously differentiable … Read more
We develop a new method called \emph{affine FR} for recovering Slater’s condition for semidefinite programming (SDP) relaxations of combinatorial optimization (CO) problems. Affine FR is a user-friendly method, as it is fully automatic and only requires a description of the problem. We provide a rigorous analysis of differences between affine FR and the existing methods. … Read more
The convex-concave minimax problem, also known as the saddle-point problem, has been extensively studied from various aspects including the algorithm design, convergence condition and complexity. In this paper, we propose a generalized asymmetric forward-backward-adjoint algorithm (G-AFBA) to solve such a problem by utilizing both the proximal techniques and the extrapolation of primal-dual updates. Besides applying … Read more
We consider network-based decentralized optimization problems, where each node in the network possesses a local function and the objective is to collectively attain a consensus solution that minimizes the sum of all the local functions. A major challenge in decentralized optimization is the reliance on communication which remains a considerable bottleneck in many applications. To … Read more
We are concerned with Lipschitzian error bounds and Lipschitzian stability properties for solutions of a complementarity system. For this purpose, we deal with a nonsmooth slack-variable reformulation of the complementarity system, and study conditions under which the reformulation serves as a local error bound for the solution set of the complementarity system. We also discuss … Read more
While the problem of tuning the hyperparameters of a support vector machine (SVM) via cross-validation is easily understood as a bilevel optimization problem, so far, the corresponding literature has mainly focused on the linear-kernel case. In this paper, we establish a theoretical framework for the development of bilevel optimization-based methods for tuning the hyperparameters of … Read more
We investigate the information complexity of mixed-integer convex optimization under different types of oracles. We establish new lower bounds for the standard first-order oracle, improving upon the previous best known lower bound. This leaves only a lower order linear term (in the dimension) as the gap between the lower and upper bounds. This is derived … Read more
Bilevel optimization problems are hierarchical problems with a constraint set which is a subset of the graph of the solution set mapping of a second optimization problem. To investigate their properties and derive solution algorithms, their transformation into single-level ones is necessary. For this, various approaches have been developed. The rst and most often used … Read more
For multistage stochastic programming problems with stagewise independent uncertainty, dynamic programming algorithms calculate polyhedral approximations for the value functions at each stage. The SDDP algorithm provides piecewise linear lower bounds, in the spirit of the L-shaped algorithm, and corresponding upper bounds took a longer time to appear. One strategy uses the primal dynamic programming recursion … Read more