A Note on Convergence Analysis of an SQP-Type Method for Nonlinear Semidefinite Programming
© Yun Wang et al. 2008
Received: 29 August 2007
Accepted: 23 November 2007
Published: 3 December 2007
We reinvestigate the convergence properties of the SQP-type method for solving nonlinear semidefinite programming problems studied by Correa and Ramirez (2004). We prove, under the strong second-order sufficient condition with the sigma term, that the local SQP-type method is quadratically convergent and the line search SQP-type method is globally convergent.
Fares et al. (2002)  studied robust control problems via sequential semidefinite programming technique. They obtained the local quadratic convergence rate of the proposed SQP-type method and employed a partial augmented Lagrangian method to deal with the problems addressed there. Correa and Ramirez (2004)  systematically studied an SQP-type method for solving nonlinear SDP problems and analyzed the convergence properties, they obtained the global convergence and local quadratic convergence rate. Both papers used the same subproblems to generate search directions, but employed different merit functions for line search. The convergence analysis of both papers depends on a set of second-order conditions without sigma term, which is stronger than no gap second-order optimality condition with sigma term.
Comparing with the work by Correa and Ramirez (2004) , in this note, we make some modifications to the convergence analysis, and prove that all results in  still hold under the strong second-order sufficient condition with the sigma term.
It should be pointed out that the importance of exploring numerical methods for solving nonlinear semidefinite programming problems has been recognized in the optimization community. For instance, Kočvara and Stingl [3, 4] have developed PENNLP and PENBMI codes for nonlinear semidefinite programming and semidefinite programming with bilinear matrix inequality constraints, respectively. Recently, Sun et al. (2007)  considered the rate of convergence of the classical augmented Lagrangian method and Noll (2007)  investigated the convergence properties of a class of nonlinear Lagrangian methods.
In Section 2, we introduce preliminaries including differential properties of the metric projector onto and optimality conditions for problem (1.1). In Section 3, we prove, under the strong second-order sufficient condition with the sigma term, that the local SQP-type method has the quadratic convergence rate and the global algorithm with line search is convergence.
2.1. Semismoothness of the Metric Projector
In this subsection, let , , and be three arbitrary finite-dimensional real spaces with a scalar product and its norm . We introduce some properties of the metric projector, especially its strong semismoothness.
The next lemma is about the generalized Jacobian for composite functions, proposed in .
2.2. Optimality Conditions
We state the strong second-order sufficient condition (SSOSC) coming from .
The next proposition relates the SSOSC and nondegeneracy condition to nonsingularity of Clarke's Jacobian of the mapping defined by (2.11). The details of this proof can be found in .
3. Convergence Analysis of the SQP-Type Method
In this section, we analyze the local quadratic convergence rate of an SQP-type method and then prove that the SQP-type method proposed in  is globally convergent. The analysis is based on the strong second-order sufficient condition, which is weaker than the conditions used in [1, 2].
3.1. Local Convergence Rate
From item (f) of [7, Theorem 4.1], we obtain the error between and directly.
Now we are in a position to state that the sequence of primal-dual points generated by Algorithm 3.1 has quadratic convergence rate.
Suppose that are twice continuously differentiable and their derivatives are locally Lipschitz in a neighborhood of a local solution to (1.1). Suppose nondegeneracy condition (2.12) and SSOSC (2.14) hold at . Consider Algorithm 3.1, in which is a minimum norm stationary point of the tangential quadratic problem (3.1). Then there exists a neighborhood of such that, if , Algorithm 3.1 is well defined and the sequence converges quadratically to .
3.2. The Global Convergence
where is the smallest eigenvalue of , denote and is a positive constant. The following proposition comes from  directly.
Now we are in a position to state the global convergence of the line search SQP Algorithm 3.7, whose proof can be found in .
Suppose that , , are continuously differentiable and their derivatives are Lipschitz continuous. Consider Algorithm 3.7, if positive definite matrices and are bounded, then one of the following situations occurs:
The research is supported by the National Natural Science Foundation of China under Project no. 10771026 and by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China.
- Fares B, Noll D, Apkarian P: Robust control via sequential semidefinite programming. SIAM Journal on Control and Optimization 2002,40(6):1791–1820. 10.1137/S0363012900373483MATHMathSciNetView ArticleGoogle Scholar
- Correa R, Ramirez HC: A global algorithm for nonlinear semidefinite programming. SIAM Journal on Optimization 2004,15(1):303–318. 10.1137/S1052623402417298MATHMathSciNetView ArticleGoogle Scholar
- Kočvara M, Stingl M: Pennon: a code for convex nonlinear and semidefinite programming. Optimization Methods & Software 2003,18(3):317–333. 10.1080/1055678031000098773MATHMathSciNetView ArticleGoogle Scholar
- Kočvara M, Stingl M: Solving nonconvex SDP problems of structural optimization with stability control. Optimization Methods & Software 2004,19(5):595–609. 10.1080/10556780410001682844MATHMathSciNetView ArticleGoogle Scholar
- Sun D, Sun J, Zhang L: The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming. Mathematical Programming 2008.Google Scholar
- Noll D: Local convergence of an augmented Lagrangian method for matrix inequality constrained programming. Optimization Methods and Software 2007,22(5):777–802. 10.1080/10556780701223970MATHMathSciNetView ArticleGoogle Scholar
- Sun D: The strong second-order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications. Mathematics of Operations Research 2006,31(4):761–776. 10.1287/moor.1060.0195MATHMathSciNetView ArticleGoogle Scholar
- Mifflin R: Semismooth and semiconvex functions in constrained optimization. SIAM Journal on Control and Optimization 1977,15(6):959–972. 10.1137/0315061MATHMathSciNetView ArticleGoogle Scholar
- Qi LQ, Sun J: A nonsmooth version of Newton's method. Mathematical Programming 1993,58(1–3):353–367.MATHMathSciNetView ArticleGoogle Scholar
- Zarantonello EH: Projections on convex sets in Hilbert space and spectral theory—I: projections on convex sets. In Contributions to Nonlinear Functional Analysis (Proceedings of a Symposium held at the Mathematics Research Center, University of Wisconsin, Madison, Wis., 1971). Edited by: Zarantonello EH. Academic Press, New York, NY, USA; 1971:237–341.Google Scholar
- Zarantonello EH: Projections on convex sets in Hilbert space and spectral theory—II: spectral theory. In Contributions to Nonlinear Functional Analysis (Proceedings of a Symposium held at the Mathematics Research Center, University of Wisconsin, Madison, Wis., 1971). Edited by: Zarantonello EH. Academic Press, New York, NY, USA; 1971:343–424.Google Scholar
- Sun D, Sun J: Semismooth matrix-valued functions. Mathematics of Operations Research 2002,27(1):150–169. 10.1287/moor.184.108.40.2062MATHMathSciNetView ArticleGoogle Scholar
- Bonnans JF, Shapiro A: Perturbation Analysis of Optimization Problems, Springer Series in Operations Research. Springer, New York, NY, USA; 2000:xviii+601.View ArticleMATHGoogle Scholar
- Bonnans JF, Gilbert JC, Lemaréchal C, Sagastizábal CA: Numerical Optimization. Theoretical and Practical Aspects, Universitext. Springer, Berlin, Germany; 2003:xiv+419.MATHGoogle Scholar
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