On implementation of a self-dual embedding method for convex programming

Abstract

In this paper, we implement Zhang's method [22], which transforms a general convex optimization problem with smooth convex constraints into a convex conic optimization problem and then apply the techniques of self-dual embedding and central path following for solving the resulting conic optimization model. A crucial advantage of the approach is that no initial solution is required, and the method is particularly suitable when the feasibility status of the problem is unknown. In our implementation, we use a merit function approach proposed by Andersen and Ye [1] to determine the step size along the search direction. We evaluate the efficiency of the proposed algorithm by observing its performance on some test problems, which include logarithmic functions, exponential functions and quadratic functions in the constraints. Furthermore, we consider in particular the geometric programming and L p -programming problems. Numerical results of our algorithm on these classes of optimization problems are reported. We conclude that the algorithm is stable, efficient and easy-to-use in general. As the method allows the user to freely select the initial solution if he/she so wishes, it is natural to take advantage of this and apply the so-called warm-start strategy, whenever the data of a new problem is not too much different from a previously solved problem. This strategy turns out to be effective, according to our numerical experience.

Semi-definite programming and quantum information

“…A standard choice for an SDP solver among the NPA community (see section 4.6) using Matlab seems to be the SeDuMi solver [291,292] created by J F Sturm, currently developed and maintained by Imre Pólik and Oleksandr Romanko under the direction of Tamás Terlaky [290]. This solver implements self-dual embedding IPM [66] and was used for instance in [226][227][228] and other works implementing NPA. Another SDP solver of particular interest in Matlab is SDPT3 solver [310,312] implemented by Toh, Todd, and Tütüncü.…”

“…In the literature there exist a couple of equivalent formulations of SDPs, each has both primal and dual forms. The author prefers the so-called standard or canonical form of SDP given below in ( 80) and ( 81) in section 3.2.1, and used in many of the classical textbooks [15,36,116], reviews [231,306], SDP fundamental papers [10,216,293,305,307] and implementations [214,292,309,310,312], sometimes with slight changes in labeling [27], different notations for the Frobenius product (3), and more general form of conic formulations [66,236]. Another important formulation is the one used by Vandenberghe and Boyd [41,315], which we provide in section 3.2.2.…”