We suggest a majorization-minimization method for solving nonconvex minimization problems. The method is based on minimizing at each iterate a properly constructed consistent majorizer of the objective function. We describe a variety of classes of functions for which such a construction is possible. We introduce an inexact variant of the method, in which only approximate minimization of the consistent majorizer is performed at each iteration. Both the exact and the inexact algorithms are shown to be descent methods whose accumulation points have a property which is stronger than standard stationarity. We give examples of cases in which the exact method can be applied. Finally, we show that the inexact method can be applied to a specific problem, called sparse source localization, by utilizing a fast optimization method on a smooth convex dual of its subproblems.

We suggest a branch and bound algorithm for solving continuous optimization problems where a (generally nonconvex) objective function is to be minimized under nonconvex inequality constraints which satisfy some specific solvability assumptions. The assumptions hold for some special cases of nonconvex quadratic optimization problems. We show how the algorithm can be applied to the problem of minimizing a nonconvex quadratic function under ball, out-of-ball and linear constraints. The main tool we utilize is the ability to solve in polynomial computation time the minimization of a general quadratic under one Euclidean sphere constraint, namely the so-called trust region subproblem , including the computation of all local minimizers of that problem. Application of the algorithm on sparse source localization problems is presented.

We consider the problem of locating a user's position from a set of noisy pseudoranges to a group of satellites. We consider both the nonlinear least squares formulation of the problem, which is nonconvex and nonsmooth, and the nonlinear squared least squares variant, in which the objective function is smooth, but still nonconvex. We show that the squared least squares problem can be reformulated as a generalized trust region subproblem and as such can be solved efficiently. Conditions for attainment of the optimal solutions of both problems are derived. The nonlinear least squares problem is shown to have tight connections to the well-known geometric circle fitting and orthogonal regression problems. Finally, a fixed point method for the nonlinear least squares formulation is derived and analyzed.