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This paper is a counterpart of Bi et al., 2011. For a locally optimal solution to the nonlinear second-order cone programming (SOCP), specifically, under Robinson’s constraint qualification, we establish the equivalence among the following three conditions: the nonsingularity of Clarke’s Jacobian of Fischer-Burmeister (FB) nonsmooth system for the Karush-Kuhn-Tucker conditions, the strong second-order sufficient condition and constraint nondegeneracy, and the strong regularity of the Karush-Kuhn-Tucker point.

The nonlinear second-order cone programming (SOCP) problem can be stated as

Let

The most popular SOC complementarity functions include the vector-valued natural residual (NR) function and Fischer-Burmeister (FB) function, respectively, defined as

Recently, with the help of [

In this work, for a locally optimal solution to the nonlinear SOCP (

Throughout this paper,

First we recall from [

The Jordan product of

Unlike scalar or matrix multiplication, the Jordan product is not associative in general. The identity element under this product is

The determinant of a vector

By the formula of spectral factorization, it is easy to compute that the projection of

The spectral factorization of the vectors

For any

If

If

The following lemma states a result for the arrow matrices associated with

For any given

Let

The following two lemmas state the properties of

For any

For any

If

If

(a) The result is direct by the equalities of Lemma

(b) Since

When

For any given

Since

Next we recall from [

We say that

To close this section, we recall from [

Unless otherwise stated, in the rest of this paper, for any

The function

For any given

If

If

If

Part (a) is immediate by noting that

(c.1):

(c.2):

Finally, we show that, when

As a consequence of Proposition

(a) The “if” part is direct by [

(b) Since

For any given

For any given

Let

Conversely, let

Now we may prove the equivalence between the

For any given

The result is direct by Proposition

Using Lemma

For any given

We first make simplifications for the last two terms in (

Let

To close this section, we establish a relation between the

Let

Since

This section studies the nonsingularity of Clarke's Jacobian of

First of all, let us take a careful look at the properties of the elements in

For

when

when

The following proposition plays a key role in achieving the main result of this section.

For

Throughout the proof, let

We next prove the implication in (

The following lemma states an important property for the elements of Clarke's Jacobian of

For any given

Since

For

Fix any

Before stating the main result of this section, we also need to recall several concepts, including constraint nondegeneracy, Robinson's constraint qualification (CQ) (see [

A feasible vector

Robinson's CQ is said to hold at a feasible solution

Clearly, the constraint nondegenerate condition (

Let

Let

Now we are in a position to prove the nonsingularity of Clarke's Jacobian of

Let

Since the nondegeneracy condition (

Note that

Let

The strong second-order sufficient condition in Definition

Any element in

Any element in

First, Lemma

In this paper, for a locally optimal solution of the nonlinear SOCP, we established the equivalence between the nonsingularity of Clarke's Jacobian of the FB system and the strong regularity of the corresponding KKT point. This provides a new characterization for the strong regularity of the nonlinear SOCPs and extends the result of [

For any given

We first prove that

In what follows, we show that