We consider quadratic programs with a single negative eigenvalue (QP1NE) subject to linear and conic constraints. The QP1NE model covers the classical clique problem that is known to be NP-hard [26]. In this paper, by combining successive linear conic optimization (SLCO), convex relaxation and line search technique, we present a global algorithm for QP1NE which can locate a global optimal solut...

Facial reduction heuristics are developed in the interest of added performance and reliability in methods for mixed-integer conic optimization. Specifically, the process of branch-and-bound is shown to spawn subproblems for which the conic relaxations are difficult to solve, and the objective bounds of linear relaxations are arbitrarily weak. While facial reduction algorithms already exist to d...

Received: date / Revised version: date Abstract. We investigate the relation between interior-point algorithms applied to two homogeneous self-dual approaches to linear programming, one of which was proposed by Ye, Todd, and Mizuno and the other by Nesterov, Todd, and Ye. We obtain only a partial equivalence of path-following methods (the centering parameter for the first approach needs to be e...

Conic optimization is the minimization of a differentiable convex objective function subject to conic constraints. We propose novel primal–dual first-order method for optimization, named proportional–integral projected gradient (PIPG). PIPG ensures that both gap and constraint violation converge zero at rate O(1/k), where k number iterations. If strongly convex, improves convergence O(1/k2). Fu...

Motivated by recent applications in robust optimization and in sign-constrained linearquadratic control, we study in this paper a new class of optimization problems called separated continuous conic programming (SCCP). Focusing on a symmetric primal-dual pair, we develop a strong duality theory for the SCCP. Our idea is to use discretization to connect the SCCP and its dual to two ordinary coni...

in this paper, we consider a stochastic time-cost tradeoff problem (tctp) in pert networks for project management, in which all activities are subjected to a linear cost function and assumed to be exponentially distributed. the aim of this problem is to maximize the project completion probability with a pre-known deadline to a predefined probability such that the required additional cost is min...

We propose a pivotal method for estimating high-dimensional sparse linear regression models, where the overall number of regressors p is large, possibly much larger than n, but only s regressors are significant. The method is a modification of the lasso, called the square-root lasso. The method is pivotal in that it neither relies on the knowledge of the standard deviation σ or nor does it need...

In this paper we investigate families of quadrics that have fixed intersections with two given hyperplanes. The cases when the two hyperplanes are parallel and when they are nonparallel are discussed. We show that these families can be described with only one parameter. In particular we show how the quadrics are transformed as the parameter changes. This research was motivated by an application...

In this paper, under a suitable regularity condition, we establish that a broad class of conic convex polynomial optimization problems, called conic sum-of-squares convex polynomial programs, exhibits exact conic programming relaxation, which can be solved by various numerical methods such as interior point methods. By considering a general convex cone-program, we give unified results that appl...