Primal-dual path following method for nonlinear semi-infinite programs with semi-definite constraints

Abstract

In this paper, we propose two algorithms for nonlinear semi-infinite semi-definite programs with infinitely many convex inequality constraints, called SISDP for short. A straightforward approach to the SISDP is to use classical methods for semi-infinite programs such as discretization and exchange methods and solve a sequence of (nonlinear) semi-definite programs (SDPs). However, it is often too demanding to find exact solutions of SDPs.Our first approach does not rely on solving SDPs but on approximately following a path leading to a solution, which is formed on the intersection of the semi-infinite region and the interior of the semi-definite region. We show weak* convergence of this method to a Karush-Kuhn-Tucker point of the SISDP under some mild assumptions and further provide with sufficient conditions for strong convergence. Moreover, as the second method, to achieve fast local convergence, we integrate a two-step sequential quadratic programming method equipped with Monteiro-Zhang scaling technique into the first method. We particularly prove two-step superlinear convergence of the second method using Alizadeh-Hareberly-Overton-like, Nesterov-Todd, and Helmberg-Rendle-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro scaling directions. Finally, we conduct some numerical experiments to demonstrate the efficiency of the proposed method through comparison with a discretization method that solves SDPs obtained by finite relaxation of the SISDP.

Local convergence of primal-dual interior point methods for nonlinear semi-definite optimization using the family of Monteiro-Tsuchiya directions

“…ℓ ). Error bounds and uniqueness of solutions of the Newton equations play key roles in the local convergence analysis of the PDIPMs using the MZ family [27,39,41], in which the proofs are based on the nonsingularity of the Jacobian of the function Φ 2 0 . However, as explained in Remark 4.1, the Jacobian of Ξ P 0 is unavailable at the KKT point w * in general and thus a different approach is necessary for proving Proposition 4.3.…”

“…Henceforth, we assume that the functions f , G, and h are three times continuously differentiable, and let w * := (x * , Y * , z * ) be a KKT point of NSDP (1.1) that satisfies the following regularity conditions. These conditions are all adopted in the recent researches [27,39,41] on PDIPMs for NSDPs:…”

“…Further examples are: the strong second-order conditions by Sun [33], sequential optimality conditions by Andreani et al [2], and the local duality by Qi [29]. Along with such theoretical results, various algorithms have been proposed for solving the NSDP, for example, augmented Lagrangian methods [2,9,11,17,34,35], sequential linear semi-definite programming methods [14], sequential quadratic semi-definite programming methods [5,8,38,43], sequential quadratically constrained quadratic semi-definite programming methods [3], interior point-type methods [12,15,20,27,28,[39][40][41][42], and so forth.…”

“…Yamakawa and Yamashita [40] introduced the shifted barrier KKT conditions as an alternative to the barrier KKT conditions for the NSDP, and in [39] they also showed the two-step superlinear convergence of the PDIPM which uses two kinds of search directions produced by solving two scaled Newton equations sharing the same Jacobian. Okuno and Fukushima [27,28] considered special NSDPs which posses an infinite number of convex inequality constraints, and proposed the PDIPM coupled with sequential quadratic programming methods.…”

“…In this paper, the local superlinear convergence of such a PDIPM is of our interest, and we leave the global convergence to future studies. The local convergence analysis is specialized to the MT family and different from the existing analysis of the PDIPMs [27,28,39,41] using the MZ scaling directions. For details, see Remark 4.1.…”