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

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.

1. Introduction

We consider the following nonlinear semidefinite programming:

(1.1)

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.

For a given matrix , its spectral decomposition is

(2.1)

where is the diagonal matrix of eigenvalues of and is a corresponding orthogonal matrix. We can express and as

(2.2)

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.

Let be a continuously differentiable function on an open neighborhood of and let be a locally Lipschitz continuous function on the open set containing . Suppose that is directionally differentiable at every point in and that is onto. Then it holds that

(2.3)

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 ;

(ii)for any and with ,

(2.4)

Furthermore, is said to be strongly semismooth at if is semismooth at and for any and with ,

(2.5)

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.

Suppose , then it has the spectral decomposition as (2.1), then the merit projector of to is denoted by and

(2.6)

where .

For our discussion, we know from [12] that the projection operator is directionally differentiable everywhere in and is a strongly semismooth matrix-valued function. In fact, for any , , there exists , satisfying

(2.7)

2.2. Optimality Conditions

Let the Lagrangian function of (1.1) be

(2.8)

Robinson's constraint qualification(CQ) is said to hold at a feasible point if

(2.9)

If is a locally optimal solution to (1.1) and Robinson's CQ holds at , then there exist Lagrangian multipliers , such that the following KKT condition holds:

(2.10)

which is equivalent to , where

(2.11)

Let be the set of all the Lagrangian multipliers satisfying (2.10). Then is a nonempty, compact convex set of if and only if Robinson's CQ holds at , see [13]. Moreover, it follows from [13] that the constraint nondegeneracy condition is a sufficient condition for Robinson constraint qualification. In the setting of the problem (1.1), the constraint nondegeneracy condition holding at a feasible point can be expressed as

(2.12)

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.

For a KKT point of (1.1), without loss of generality, we assume that and have the spectral decomposition forms

(2.13)

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

Definition 2.3.

Let be a stationary point of (1.1) such that (2.12) holds at . One says that the strong second-order sufficient condition holds at if

(2.14)

where , is the affine hull of the critical cone :

(2.15)

And the linear-quadratic function is defined by

(2.16)

is the Moore-Penrose pseudoinverse of .

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

Linearizing (1.1) at the current point , we obtain the following tangent quadratic problem:

(3.1)

where . Let be a KKT point of (3.1), then we have , where

(3.2)

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.

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 . Then there exists a neighborhood of such that if in , (3.1) has a local solution together with corresponding Lagrangian multiplies satisfying

(3.3)

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.

By Theorem 3.2, we know Algorithm 3.1 is well defined. Let

(3.4)

then

(3.5)

where is the minimum norm solution to (3.1), and , are the associated multipliers. Using Taylor expansion of (3.2) at , noting that , , and (3.5), we obtain

(3.6)

As the projection operator is strongly semismooth, we have that there exists such that

(3.7)

Since

(3.8)

we have

(3.9)

Noting the fact that , by Taylor expansion of the third equation of (3.2) at , we obtain

(3.10)

Therefore, we can conclude that

(3.11)

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

The tangential quadratic problem constrained here is slightly more general than (3.1) in the sense that the Hessian of the Lagrangian is replaced by some positive definite matrix . Thus the tangential quadratic problem in now becomes

(3.12)

The KKT systemof (3.12) is

(3.13)

To obtain theglobal convergence, we use the Han penalty function given by [14], as a merit function and Armijo line search. For problem (1.1), the Han penalty function is defined by

(3.14)

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

Proposition 3.4.

• (i) If , , have a directional derivative at in the direction , then has also a directional derivative at in the direction . If, in addition, is feasible for (1.1), we have

  

  (3.15)

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.

Suppose nondegeneracy condition (2.12) and SSOSC (2.14) hold at . Then there exists , such that for any there exist a neighborhood of and a neighborhood of , for any , the problem

(3.16)

has a unique solution denote . The function is locally Lipschitz continuous and semismooth on . Furthermore, there exists , for any ,

(3.17)

where

(3.18)

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.

For the definition of the projection operator , we have

(3.19)

and for any , ,

(3.20)

Then

(3.21)

holds for any . So taking and , we obtain that

(3.22)

which implies

(3.23)

Since , for any fixed , there exists a neighborhood of such that

(3.24)

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.

Compute

(3.25)

Using backtracking line search rule to compute the step length :

Step 6.

set , ;

Step 7.

if

(3.26)

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 .