Fat conic section and fat conic spline are defined. With well established properties of fat conic splines, the problem of approximating a ruled surface by a tangent smooth cone spline can then be changed as the problem of fitting a plane fat curve by a fat conic spline. Moreover, the fitting error between the ruled surface and the cone spline can be estimated explicitly via fat conic spline fit...

Gauge functions significantly generalize the notion of a norm, and gauge optimization, as defined by Freund (1987), seeks the element of a convex set that is minimal with respect to a gauge function. This conceptually simple problem can be used to model a remarkable array of useful problems, including a special case of conic optimization, and related problems that arise in machine learning and ...

We consider linear optimization problems over the cone of copositive matrices. Such conic optimization problems, called copositive programs, arise from the reformulation of a wide variety of difficult optimization problems. We propose a hierarchy of increasingly better outer polyhedral approximations to the copositive cone. We establish that the sequence of approximations is exact in the limit....

In this paper we consider l0 regularized convex cone programming problems. In particular, we first propose an iterative hard thresholding (IHT) method and its variant for solving l0 regularized box constrained convex programming. We show that the sequence generated by these methods converges to a local minimizer. Also, we establish the iteration complexity of the IHT method for finding an -loca...

Conic programming has been lately one of the most dynamic area of the optimization field. Although a lot of attention was focused on designing and analyzing interior-point algorithms for solving optimization problems, the class of analytic center cutting plane methods was less investigated. These methods are designed to solve feasibility problems by finding points which are interior to differen...

We propose a framework for modeling and solving low-rank optimization problems to certifiable optimality. introduce symmetric projection matrices that satisfy $Y^2=Y$, the matrix analog of binary variables $z^2=z$, model rank constraints. By leveraging regularization strong duality, we prove this paradigm yields tractable convex over non-convex set orthogonal matrices. Furthermore, design outer...

We consider penalized formulations of machine learning problems with regularization penalty having conic structure. For several important learning problems, state-of-the-art optimization approaches such as proximal gradient algorithms are difficult to apply and computationally expensive, preventing from using them for large-scale learning purpose. We present a conditional gradient algorithm, wi...

We study two issues on condition numbers for convex programs: one has to do with the growth of the condition numbers of the linear equations arising in interior-point algorithms; the other deals with solving conic systems and estimating their distance to infeasibility. These two issues share a common ground: the key tool for their development is a simple, novel perspective based on implicitly-d...

The main purpose of this paper consists of providing characterizations of the inclusion of the solution set of a given conic system posed in a real locally convex topological space into a variety of subsets of the same space de…ned by means of vector-valued functions. These Farkas-type results are used to derive characterizations of the weak solutions of vector optimization problems (including ...

We develop a natural variant of Dikin’s affine-scaling method, first for semidefinite programming and then for hyperbolic programming in general. We match the best complexity bounds known for interior-point methods. All previous polynomial-time affine-scaling algorithms have been for conic optimization problems in which the underlying cone is symmetric. Hyperbolicity cones, however, need not be...