We make a unified analysis of interior proximal methods of solving convex second-order cone programming problems. These methods use a proximal distance with respect to second-order cones which can be produced with an appropriate closed proper univariate function in three ways. Under some mild conditions, the sequence generated is bounded with each limit point being a solution, and global rates of convergence estimates are obtained in terms of objective values. A class of regularized proximal distances is also constructed which can guarantee the global convergence of the sequence to an optimal solution. These results are illustrated with some examples. In addition, we also study the central paths associated with these distance-like functions, and for the linear SOCP we discuss their relations with the sequence generated by the interior proximal methods. From this, we obtain improved convergence

We introduce an entropy-like proximal algorithm for the problem of minimizing a closed proper convex function subject to symmetric cone constraints. The algorithm is based on a distance-like function that is an extension of the Kullback-Leiber relative entropy to the setting of symmetric cones. Like the proximal algorithms for convex programming with nonnegative orthant cone constraints, we show that, under some mild assumptions, the sequence generated by the proposed algorithm is bounded and every accumulation point is a solution of the considered problem. In addition, we also present a dual application of the proposed algorithm to the symmetric cone linear program, leading to a multiplier method which is shown to possess similar properties as the exponential multiplier method (Tseng and Bertsekas in Math. Program. 60:119, 1993) holds.

In this paper, we study the properties of the penalized Fischer-Burmeister (FB) second-order cone (SOC) complementarity function. We show that the function possesses similar desirable properties of the FB SOC complementarity function for local convergence; for example, with the function the second-order cone complementarity problem (SOCCP) can be reformulated as a (strongly) semismooth system of equations, and the corresponding nonsmooth Newton method has local quadratic convergence without strict complementarity of solutions. In addition, the penalized FB merit function has bounded level sets under a rather weak condition which can be satisfied by strictly feasible monotone SOCCPs or SOCCPs with the Cartesian <i>R</i> ₀₁-property, although it is not continuously differentiable. Numerical results are included to illustrate the theoretical considerations.

We provide an affirmative answer to an question that the Fischer-Burmeister complementarity function associated with symmetric cones, named the FB SC complementarity function, is globally Lipschitz continuous and strongly semismooth everywhere for Hn and Qn. This is achieved with the help of embedding Hn and Qn into certain Sm.

This work combines a level-set approach and the optimal transport-based Wasserstein distance in a data assimilation framework. The primary motivation of this work is to reduce assimilation artifacts resulting from the position and observation error in the tracking and forecast of pollutants present on the surface of oceans or lakes. Both errors lead to spurious effect on the forecast that need to be corrected. In general, the geometric contour of such pollution can be retrieved from observation while more detailed characteristics such as concentration remain unknown. Herein, level sets are tools of choice to model such contours and the dynamical evolution of their topology structures. They are compared with contours extracted from observation using the Wasserstein distance. This allows to better capture position mismatches between both sources compared with the more classical Euclidean distance. Finally, the viability of this approach is demonstrated through academic test cases and its numerical performance is discussed.

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We consider a class of derivative-free descent methods for solving the second-order cone complementarity problem (SOCCP). The algorithm is based on the FischerBurmeister (FB) unconstrained minimization reformulation of the SOCCP, and utilizes a convex combination of the negative partial gradients of the FB merit function _FB as the search direction. We establish the global convergence results of the algorithm under monotonicity and the uniform Jordan <i>P</i>-property, and show that under strong monotonicity the merit function value sequence generated converges at a linear rate to zero. Particularly, the rate of convergence is dependent on the structure of second-order cones. Numerical comparisons are also made with the limited BFGS method used by Chen and Tseng (<i>An unconstrained smooth minimization reformulation of the second-order cone complementarity problem</i>, Math. Program. 104(2005), pp

Necessary conditions of optimality are derived for multiobjective optimal control problems with free end-time, in which the dynamics constraint is modeled as a nonconvex differential inclusion. The obtained results cover some previous results on necessary conditions for multiojective and single objective optimal control problems.

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In this paper, we study the existence of local and global saddle points for nonlinear second-order cone programming problems. The existence of local saddle points is developed by using the second-order sufficient conditions, in which a sigma-term is added to reflect the curvature of second-order cone. Furthermore, by dealing with the perturbation of the primal problem, we establish the existence of global saddle points, which can be applicable for the case of multiple optimal solutions. The close relationship between global saddle points and exact penalty representations are discussed as well.