The primal-dual Dikin-type affine scaling method was originally proposed for linear optimization and then extended to semidefinite optimization. Here, the method is generalized to symmetric conic optimization using the notion of Euclidean Jordan algebras. The method starts with an interior feasible but not necessarily centered primal-dual solution, and it features both centering and reducing th...

Robustness is a fundamental issue in signal processing; unmodeled dynamics and unexpected noise in systems and signals are inevitable in designing systems and signals. Against such uncertainties, min-max optimization, or worst case optimization is a powerful tool. In this light, we propose an efficient design method of FIR (finite impulse response) digital filters for approximating and invertin...

A robust structural optimization scheme as well as an optimization algorithm are presented based on the robustness function. Under the uncertainties of external forces based on the info-gap model, the maximization problem of the robustness function is formulated as an optimization problem with infinitely many constraints. By using the quadratic embedding technique of uncertainty and the S-proce...

In recent years, optimization theory has been greatly impacted by the advent of sum of squares (SOS) optimization. The reliance of this technique on large-scale semidefinite programs however, has limited the scale of problems to which it can be applied. In this paper, we introduce DSOS and SDSOS optimization as more tractable alternatives to sum of squares optimization that rely instead on line...

This paper deals with semidefinite bounds for the k-cluster problem, a classical NPhard problem in combinatorial optimization. We present numerical experiments to compare the standard semidefinite bound with the new semidefinite bound of [MR09], regarding the trade-off between tightness and computing time. We show that the formulation of the semidefinite bounds has an impact on the efficiency o...

ABSTRACT We describe efficient interior-point methods for the design of filters with constraints on the magnitude spectrum, for example, piecewise-constant upper and lower bounds, and arbitrary phase. Several researchers have observed that problems of this type can be solved via convex optimization and spectral factorization. The associated optimization problems are usually solved via linear pr...

We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the positive semidefinite rank, and their symmetric analogues: the completely positive rank and the completely positive semidefinite rank. We study the convergence properties of o...