- Research Article
- Open Access

# A Note on Convergence Analysis of an SQP-Type Method for Nonlinear Semidefinite Programming

- Yun Wang
^{1}Email author, - Shaowu Zhang
^{2}and - Liwei Zhang
^{1}

**2008**:218345

https://doi.org/10.1155/2008/218345

© Yun Wang et al. 2008

**Received:**29 August 2007**Accepted:**23 November 2007**Published:**3 December 2007

## Abstract

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.

## Keywords

- Line Search
- Merit Function
- Constraint Qualification
- Augmented Lagrangian Method
- Nondegeneracy Condition

## 1. Introduction

where , , , and are twice continuously differentiable functions, is the linear space of all real symmetric matrices, and is the cone of all symmetric positive semidefinite matrices.

Fares et al. (2002) [1] 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) [2] 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) [2], in this note, we make some modifications to the convergence analysis, and prove that all results in [2] 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) [5] considered the rate of convergence of the classical augmented Lagrangian method and Noll (2007) [6] 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. Preliminaries

We use to denote the set of all the matrices of rows and columns. For and in , we use the Frobenius inner product , and the Frobenius norm , where "tr" denotes the trace operation of a square matrix.

where , , are the index sets of positive, zero, negative eigenvalues of , respectively.

### 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 [7].

Lemma 2.1.

where is defined by , .

The following concept of semismoothness was first introduced by Mifflin [8] for functionals and was extended by Qi and Sun in [9] to vector valued functions.

Definition 2.2.

Let be a locally Lipschitz continuous function on the open set . One says that is semismooth at a point if

(i) is directionally differentiable at ;

Let be a closed convex set in a Banach space , and let be the metric projector over . It is well known in [10, 11] that is -differentiable almost everywhere in and for any , is well defined.

where .

### 2.2. Optimality Conditions

where is the lineality space of the tangent cone of at . If , a locally optimal solution to (1.1), is nondegenerate, then is a singleton.

We state the strong second-order sufficient condition (SSOSC) coming from [7].

Definition 2.3.

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 [7].

Proposition 2.4.

Let be a KKT point of (1.1). If nondegeneracy condition (2.12) and SSOSC (2.14) hold at , then any element in is nonsingular, where is defined by (2.11).

## 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 [2] 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

The following algorithm is an SQP-type algorithm for solving (1.1), which is based on computing at each iteration a primal-dual stationary point of (3.1).

Algorithm 3.1.

Step 1.

Given an initial iterate point . Compute , , , and . Set .

Step 2.

If , , , stop.

Step 3.

Compute , and find a solution to (3.1).

Step 4.

Set .

Step 5.

Compute , , , and . Set and go to step 2.

From item (f) of [7, Theorem 4.1], we obtain the error between and directly.

Theorem 3.2.

Now we are in a position to state that the sequence of primal-dual points generated by Algorithm 3.1 has quadratic convergence rate.

Theorem 3.3.

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 .

Proof.

Since the nondegeneracy condition (2.12) and SSOSC (2.14) hold, we have from Proposition 2.4 that (3.11) implies the quadratic convergence of the sequence .

### 3.2. The Global Convergence

where is the smallest eigenvalue of , denote and is a positive constant. The following proposition comes from [2] directly.

- (i)

where is the matrix whose columns form an orthonormal basis of .

(ii)If is a feasible point of (1.1) and has a local minimum at , then is the local solution to (1.1). Furthermore, if , , are differentiable at and nondegeneracy condition (2.12) holds at , then .

(iii)If and , then .

To discuss the conditions ensuring the exactness of , we need the following lemma from (3.10).

Lemma 3.5.

is the augmented Lagrangian function with the penalty parameter for (1.1).

Theorem 3.6.

Suppose that , , are twice differentiable around a local solution to (1.1), at which nondegeneracy condition (2.12) and SSOSC (2.14) hold. If , then has a strict local minimum at .

Proof.

From Lemma 3.5, we know that there exist an and a neighborhood of where is a strict minimum of . So we can conclude that is a strict minimum of on .

Let us outline the line-search SQP-type algorithm that uses the merit function defined in (3.14) and the parameter updating scheme from [14], which is a generalized version to the algorithm in [2].

Algorithm 3.7.

Step 1.

Given a positive number , , . Choose an initial iterate . Compute , , , , and . Set .

Step 2.

If , stop.

Step 3.

Compute a symmetric matrix and find a solution to (3.12).

Step 4.

Adapt .

if

then

else

Step 5.

Using backtracking line search rule to compute the step length :

Step 6.

set , ;

Step 7.

holds for , then and stop the line search.

Step 8.

else, choose ;

Step 9.

set , go to step 7

Step 10.

Set , , .

Step 11.

Compute and . Set and go to 2

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 [2].

Theorem 3.8.

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:

(i)the sequence is unbounded, in which case is also unbounded;

(ii)there exists an index such that for any , and one of the following situations occurs:

(a)

(b) and .

## Declarations

### Acknowledgments

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.

## Authors’ Affiliations

## References

- 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.27.1.150.342MATHMathSciNetView 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

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.