To obtain a primal-dual pair of conic programming problems having zero duality gap, two methods have been proposed: the facial reduction algorithm due to Borwein and Wolkowicz [1, 2] and the conic expansion method due to Luo, Sturm, and Zhang [5]. We establish a clear relationship between them. Our results show that although the two methods can be regarded as dual to each other, the facial redu...

A Unifying Framework for the Convexification of Mixed-Integer Conic Binary Sets The paper “Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications,” by Fatma Kilinc-Karzan, Simge Kucukyavuz, Dabeen Lee, Soroosh Shafieezadeh-Abadeh, develops a unifying framework convexifying mixed-integer conic binary sets. Many applications in machine-learning operations research give rise to in...

Unsupervised learning algorithms based on convex and conic encoders are proposed. The encoders find the closest convex or conic combination of basis vectors to the input. The learning algorithms produce basis vectors that minimize the reconstruction error of the encoders. The convex algorithm develops locally linear models of the input, while the conic algorithm discovers features. Both algorit...

In this paper, we describe a conic approach to the stability theory of uncertain polynomials. We present necessary and sufficient conditions for a conic set p0 + K of polynomials to be Hurwitz stable (K is a convex cone of polynomials of degree ≤ n and deg p0 = n). As analytical tools we derive an edge theorem and Rantzer type conditions for marginal stability (semi-stability). The results are ...

The paper considers solving of linear programming problems with p-order conic constraints that are related to a certain class of stochastic optimization models with risk objective or constraints. The proposed approach is based on construction of polyhedral approximations for p-order cones, and then invoking a Benders decomposition scheme that allows for efficient solving of the approximating pr...

We discuss natural connections between three objects: quadratic forms with values in line bundles, conic bundles and quaternion orders. We use the even Clifford algebra, and the Brauer-Severi Variety, and other constructions to give natural bijections between these objects under appropriate hypothesis. We then restrict to a surface base and we express the second Chern class of the order in term...

Inspired by the principle of satisficing (Simon 1955), Long et al. (2021) propose an alternative framework for optimization under uncertainty, which we term as a robust model. Instead sizing uncertainty set in optimization, model is specified target objective with aim delivering solution that least impacted achieving target. At heart this framework, minimize level constraint violation all possi...

Structured light is a perception method that allows us to obtain 3D information from images of the scene by projecting synthetic features with a light emitter. Traditionally, this method considers a rigid configuration, where the position and orientation of the light emitter with respect to the camera are known and calibrated beforehand. In this paper we propose a new omnidirectional structured...