On the Second Order Optimality Conditions for Optimization Problems with Inequality Constraints

A nonlinear optimization problem (P) with inequality constraints can be converted into a new optimization problem (PE) with equality constraints only. This is a Valentine method for finite dimensional optimization. We review second order optimality conditions for (PE) in connection with those of (P). A strictly complementary slackness condition can be made to get the property that sufficient optimality conditions for (P) imply the same property for (PE). We give some new results (see Theorems 3.1, 3.2 and 3.3) .Without any assumption, a counterexample is given to show that these conditions are not equivalent.


Introduction
Results on second order optimality conditions can be found in [1][2][3].
Converting inequality constraints into equality constraints, we get the following problem: This method is known to be a Valentine method [4][5][6].In the literature, second order optimality conditions for Valentine method are not studied.
Second order optimality conditions for (P) are related to copositivity and are NP-hard [7].Second order optimality conditions for (PE) are related to the definiteness of a matrix in a vector subspace and there are efficient algorithms [8].A strictly complementary slackness condition must be made to get the property that sufficient optimality conditions for (P) imply the same property for (PE).
Recall that the classical Karush-Kuhn-Tucker first and second order optimality conditions for a local minimizer * x for (P), stated under the linear independence constraint qualification (LICQ), can be written as follows: There exists a unique Lagrange multiplier *  such that the Lagrangian function: The sufficient second order optimality condition for a feasible point x is: there exists a Lagrange multiplier The fact that *  or  is the same for all critical vectors is very restrictive and, without (LICQ) or convexity assumptions, very difficult to get [9,10].Recently, many authors have weakened the constraint qualification (LICQ) and  

CN
are obtained [9][10][11].In Daldoul-Baccari [12], a numerical method is given in order to test the constant rank condition of Martinez et al. [11].Another difficulty is that there is no efficient algorithm to test the conditions: a vector subspace and efficient algorithms for   2 CN exist ( [8]).
In this paper, we are interested by the use of efficient

CS
A first step is the conversion of (P) into (PE).
Our main result is the following theorem.It is stated without any constraint qualification (linear independence, Mangasarian-Fromovitz or convexity assumptions).if and only if the following conditions hold:   is a generalized Lagrange multiplier for   satisfies sufficient second order optimality conditions for (P) at * x Note that 1) The minimizers * x and   , x y cited in the first item of Theorem 1.1 could be local or global.
2) The strict complementary slackness condition 0 and can be seen by the following simple counterexample: 3) Existence of Lagrange multiplier in the second and third item of the theorem is not guaranteed [9].
We begin with some notations:  The (generalized) Lagrangian function  of (P) is  The gradient of f with respect to x is the column vector is its transpose and the gradient of  with respect to x is the column vector is the set of generalized Lagrange multipliers of (P) at .
 The set of normalized Lagrange multipliers of (P) at * 0  Necessary first and second order optimality conditions for (P) can be written in the following form ([13], Theorem 9.3).Theorem 1.2.If * x is a local minimizer for (P) then: T xx The necessary second order optimality conditions (1.2) can be written as In the same way, for a local minimizer x y  of (PE), we have x   the generalized sufficient second order optimality conditions, , for (P) are that are not empty and, for every For 0 * X   , the generalized sufficient second order optimality conditions, are not empty and, for every  The classical sufficient second order optimality condition for (P), at * x   , is that there exists  In the same way, the classical sufficient second order optimality condition for (PE), at 0 * X   , is that there exists In [14,15], the existence of such multipliers is studied.
satisfies the sufficient second order optimality conditions for (P) if satisfies the sufficient second order optimality conditions for (PE) if

Some Properties of (PE) and (P)
Let * x be a local minimizer for (P) and * X the corresponding minimizer for (PE).It is easy to see that The following properties are easy to check: then there exists dy such that

Optimality under Regularity
We begin with the regular case and extend the result of ( [16], Proposition 1.32).Theorem.3.1.Let * x be a feasible point for (P), satisfies (LICQ) and (SCS), then the classical sufficient second order optimality condition (1.8) holds if and only if the classical sufficient second order optimality condition (1.9) holds.
Proof.* x satisfies (LICQ) and and, from (SCS), we have The first part of the theorem is the Proposition 1.32 of [16].To prove the "only if", Let In the above theorem, (LICQ) is not necessary and one can prove the following theorem (see [16], Proposition 1.31).
satisfies the sufficient second order optimality conditions for (P) if and only if it satisfies the sufficient second order optimality conditions for (PE) hold.
The main result of this section is the following theorem (compare with [16], Proposition 1.32).
Theorem.3.3.Let * x be a feasible point for (P) and such that 1) There exists a multiplier     2) The sufficient second order optimality condition x y  satisfies the sufficient second order optimality conditions and we have two cases: 1) d 0.
x  This means that d 0 y  , d 0

A Counterexample
The following counterexample shows that We list some properties of (P): 1) , g x y g x y y g x y   and Consider the associated (PE) problem: is a minimizer for (PE) and we get We can prove Lemma 4.2.
For every We conclude that we have two cases: 1) , .
i j a a j i    We have two cases a) There exists i j  such that 0 2) There exists i such that a 0 i a  and j such that 0 j a  , it is easy to find 0 i   and 5. Proof of Theorem 1.1.

1) Suppose *
x is a local minimizer for (P).There exists an open ball   .

Concluding Remarks
In the regular case and in the presence of strictly complementary slackness (SCS), we have shown that an optimization problem (P), with inequality constraints, can be converted into an optimization problem (PE) with equality constraints in such a way that sufficient second order optimality conditions are preserved.Without any regularity assumption, we have shown that sufficient second order optimality conditions hold for (PE) if these hold for (P) and if (SCS) holds.

  and this is true for any i  3 )
Suppose that   0 ,  satisfies the sufficient second order optimality conditions for (PE).This means that for allWe get that 1 0   and this is true for any   * , .i i I x  