In this paper, we study polynomial-time interior-point algorithms in view of information geometry. We introduce an information geometric structure for a conic linear program based on a self-concordant barrier function. Riemannian metric is defined with the Hessian of the barrier function. We introduce two connections $\nabla$ and $\nabla^*$ which roughly corresponds to the primal and … Read more
In this paper we deal with the Critical Node Problem (CNP), i.e., the problem of searching for a given number K of nodes in a graph G, whose removal minimizes the number of connections between pairs of nodes in the residual graph. While the NP-completeness of this problem for general graphs has been already established … Read more
We present in this paper two different classes of general $K$-splitting algorithms for solving finite-dimensional convex optimization problems. Under the assumption that the function being minimized has a Lipschitz continuous gradient, we prove that the number of iterations needed by the first class of algorithms to obtain an $\epsilon$-optimal solution is $O(1/\epsilon)$. The algorithms in … Read more
It is shown that the steepest descent and Newton’s method for unconstrained nonconvex optimization under standard assumptions may be both require a number of iterations and function evaluations arbitrarily close to O(epsilon^{-2}) to drive the norm of the gradient below epsilon. This shows that the upper bound of O(epsilon^{-2}) evaluations known for the steepest descent … Read more
In this paper we analyze the iteration-complexity of the hybrid proximal extragradient (HPE) method for finding a zero of a maximal monotone operator recently proposed by Solodov and Svaiter. One of the key points of our analysis is the use of new termination criteria based on the $\varepsilon$-enlargement of a maximal monotone operator. The advantage … Read more
Consider the following supposedly-simple problem: “compute x \in S” where S is a convex set conveyed by a separation oracle, with no further information (e.g., no bounding ball containing or intersecting S, etc.). Our interest in this problem stems from fundamental issues involving the interplay of computational complexity, the geometry of S, and the stability … Read more
We consider an optimization model for determining optimal opportunistic maintenance (that is, component replacement) schedules when data is deterministic. This problem generalizes that of Dickman, Epstein, and Wilamowsky [21] and is a natural starting point for the modelling of replacement schedules when component lives are non-deterministic. We show that this basic opportunistic replacement problem is … Read more
Branching is an important component of branch-and-bound algorithms for solving mixed integer linear programs. We consider the problem of selecting, at each iteration of the branch-and-bound algorithm, a general branching disjunction of the form $“\pi x \leq \pi_0 \vee \pi x \geq \pi_0 + 1”$, where $\pi, \pi_0$ are integral. We show that the problem … Read more
This paper considers an important class of convex programming problems whose objective function $\Psi$ is given by the summation of a smooth and non-smooth component. Further, it is assumed that the only information available for the numerical scheme to solve these problems is the subgradient of $\Psi$ contaminated by stochastic noise. Our contribution mainly consists … Read more
We focus on a permutation betting market under parimutuel call auction model where traders bet on the final ranking of n candidates. We present a Proportional Betting mechanism for this market. Our mechanism allows the traders to bet on any subset of the n x n ‘candidate-rank’ pairs, and rewards them proportionally to the number … Read more