When addressing the maximum stable set problem on a graph G = (V,E), rank inequalities prescribe that, for any subgraph G[U] induced by U ⊆ V , at most as many vertices as the stability number of G[U] can be part of a stable set of G. These inequalities are very general, as many of … Read more
In this work we present a new feasible direction algorithm for solving smooth nonlinear second-order cone programs. These problems consist of minimizing a nonlinear di erentiable objective function subject to some nonlinear second-order cone constraints. Given a point interior to the feasible set de nfined by the nonlinear constraints, the proposed approach computes a feasible and descent … Read more
We complete the complexity classification by degree of minimizing a polynomial in two variables over the integer points in a polyhedron. Previous work shows that in two variables, optimizing a quadratic polynomial over the integer points in a polyhedral region can be done in polynomial time, while optimizing a quartic polynomial in the same type … Read more
Decision rules provide a flexible toolbox for solving the computationally demanding, multistage adaptive optimization problems. There is a plethora of real-valued decision rules that are highly scalable and achieve good quality solutions. On the other hand, existing binary decision rule structures tend to produce good quality solutions at the expense of limited scalability, and are … Read more
In this paper, we study differential properties of Euclidean projection onto the power cone $K^{(p,q)}_n=\{(x,y,z)\in \mathbb{R}_+\times \mathbb{R}_+\times \mathbb{R}^n,\norm{z} \leq x^p y^q\}$, where $0< p,q < 1, p+q=1$. Projections onto certain power cones are examples of semismooth but non-strongly-semismooth projection onto a convex cone. CitationDivision of Mathematical Sciences, School of Physical & Mathematical Sciences, Nanyang Technological ... Read more
The alternating direction method of multipliers (ADMM) is widely used in solving structured convex optimization problems. Despite of its success in practice, the convergence of the standard ADMM for minimizing the sum of $N$ $(N\geq 3)$ convex functions whose variables are linked by linear constraints, has remained unclear for a very long time. Recently, Chen … Read more
The alternating direction method of multipliers (ADMM) has been widely used for solving structured convex optimization problems. In particular, the ADMM can solve convex programs that minimize the sum of $N$ convex functions with $N$-block variables linked by some linear constraints. While the convergence of the ADMM for $N=2$ was well established in the literature, … Read more
Primal-dual splitting schemes are a class of powerful algorithms that solve complicated monotone inclusions and convex optimization problems that are built from many simpler pieces. They decompose problems that are built from sums, linear compositions, and infimal convolutions of simple functions so that each simple term is processed individually via proximal mappings, gradient mappings, and … Read more
The fraction of customers who choose a particular item from among a set of available items can be increased significantly by the inclusion of a related inferior (and apparently irrelevant) item in the choice set. This violation of the independence from irrelevant alternatives and the regularity properties is called the decoy effect, dominance effect, or … Read more
We consider a class of sampling-based decomposition methods to solve risk-averse multistage stochastic convex programs. We prove a formula for the computation of the cuts necessary to build the outer linearizations of the recourse functions. This formula can be used to obtain an efficient implementation of Stochastic Dual Dynamic Programming applied to convex nonlinear problems. … Read more