We develop a natural generalization to the notion of the central path – a notion that lies at the heart of interior-point methods for convex optimization. The generalization is accomplished via the “derivative cones” of a “hyperbolicity cone,” the derivatives being direct and mathematicallyappealing relaxations of the underlying (hyperbolic) conic constraint, be it the non-negative orthant, the...

The Direct Weight Optimization (DWO) approach to estimating a regression function and its application to nonlinear system identification has been proposed and developed during the last few years by the authors. Computationally, the approach is typically reduced to a quadratic or conic programming and can be effectively realized. The obtained estimates demonstrate optimality or sub-optimality in...

We present a new solver for non-convex trajectory optimization problems that is specialized robotics applications. CALIPSO, or the Conic Augmented Lagrangian Interior-Point SOlver, combines several strategies constrained numerical to natively handle second-order cones and complementarity constraints. It reliably solves challenging motion planning include contact-implicit formulations of impacts...

This paper proposes a convex relaxation of a sparse support vector machine (SVM) based on the perspective relaxation of mixed-integer nonlinear programs. We seek to minimize the zero-norm of the hyperplane normal vector with a standard SVM hinge-loss penalty and extend our approach to a zeroone loss penalty. The relaxation that we propose is a second-order cone formulation that can be efficient...

We consider minimizing a conic quadratic objective over a polyhedron. Such problems arise in parametric value-at-risk minimization, portfolio optimization, and robust optimization with ellipsoidal objective uncertainty; and they can be solved by polynomial interior point algorithms for conic quadratic optimization. However, interior point algorithms are not well-suited for branch-and-bound algo...

Abstract Recently, Bomze et al. introduced a sparse conic relaxation of the scenario problem two stage stochastic version standard quadratic optimization problem. When compared numerically to Burer’s classical reformulation, authors showed that there seems be almost no difference in terms solution quality, whereas time can differ by orders magnitudes. While did find very limited special case, f...

Random projections can reduce the dimensionality of point sets while keeping approximate congruence. Applying random to optimization problems raises many theoretical and computational issues. Most issues in application conic programming were addressed Liberti et al. (Linear Algebr. Appl. 626:204–220, 2021) [1]. This paper focuses on semidefinite programming.

A novel model, called evolutionary patched model, based on the patched conic approximation is applied to the optimization of space missions with engineering constraints in this paper. The interplanetary trajectory consists of geocentric escape orbit, heliocentric transfer orbit and target capture orbit. The model, firstly, gets the escape orbit and capture orbit by optimizing the elements of or...

We discuss some properties of the distance to infeasibility of a conic linear system Ax = b; x 2 C; where C is a closed convex cone. Some interesting connections between the distance to infeasibility and the solution of certain optimization problems are established. Such connections provide insight into the estimation of the distance to infeasibility and the explicit computation of infeasible p...

The famous Frank–Wolfe theorem ensures attainability of the optimal value for quadratic objective functions over a (possibly unbounded) polyhedron if the feasible values are bounded. This theorem does not hold in general for conic programs where linear constraints are replaced by more general convex constraints like positive-semidefiniteness or copositivity conditions, despite the fact that the...