Constructing a continuously differentiable exact augmented Lagrangian function for nonlinear semidefinite programming (The state-of-the-art optimization technique and future development)

Constructing

a

continuously

differentiable

exact

augmented

Lagrangian

function for nonlinear semidefinite

_programming

Ellen Hidemi Fukuda*

Graduate School

_{of Informatics, Kyoto University}

Bruno

_{Figueira Lourengo}

Faculty

of

Science and

_Technology,

Seikei

_University

Abstract

In this_note,wewillconstructa_continuoulydifferentiableexact_{augmented Lagrangian}

function for nonlinear semidefinite_{programming problems.} Thisfunction is definedon

the_productspaceof the_problem’svariables and of the_multipliers. The unconstrained

minimizationof theproposedexactaugmented

Lagrangian

givesasolution of theorig‐

inal

_problem,

when the

_penalty

_parameter is _{large enough.} We will show that the

exactness_propertyholds when the

_{nondegeneracy}

condition is assumed.

Keywords:

Nonlinear semidefinite_programming, exact_{augmented Lagrangian} func‐

tions,exact

_penalty

functions,

nondegeneracy.

1

Introduction

The

_following

nonlinear

_{semidefinite programming}

_(NSDP)

_problem

is considered: minimize

_f(x)

x

_(NSDP)

subject

to

_{G(x)\in \mathrm{S}_{+}^{m},}

where

_{f:\mathbb{R}^{n}\rightarrow \mathbb{R}}

and G:\mathbb{R}^{n}\rightarrow \mathrm{S}^{m} are twice

continuously

differentiable

functions,

\mathrm{S}^{m} is

the linear_spaceof all real

_symmetric

matricesof dimensionm\times m,and

\mathrm{S}_{+}^{m}

istheconeof all

positive

semidefinite matricesin\mathrm{S}^{m}.

Here,

weomit

equality

constraints

just

for

simplicity.

The above formulation is

considerably

new, but it was

already

used in many

application

fields,

like control

_theory

_{[2, 12],}

structural

_optimization

_{[16, 18],}

truss

_design

_problems

_[3],

and finance

_[17].

The research associated to NSDP

_is,

_{however, relatively}

_scarce, ifwe

compare tothe

particular

case of linear semidefinite

programming problems.

We referto

_[22]

for a_survey of numerical methods for NSDP

problems.

In

particular,

it describes the

_{augmented Lagrangian,}

the

_{sequential quadratic programming,}

and the

primal‐dual

interior

_point

methods. In this _paper, we_propose another method for

solving

’This workwassupported byGrant‐in‐AidforYoungScientists

(B)

(26730012)from_JapanSocietyfor

general

NSDP

_problems.

More

_precisely,

we construct a

continuously

differentiable exact

augmented

Lagrangian

function for NSDP.

_By

_exact, wemeanthat anunconstrained mini‐

mization of this functionrecovers asolutionof the

_{original problem,}

whenacertain

penalty

parameteris

_{large enough.}

The exact

_{augmented Lagrangian}

function also differs from the

exact

_penalty

one. In

fact,

the former is defined on the

product

space of the

problem’s

variables and of the

_{Lagrange multipliers,}

while the latter isdefined in the same _space of

the

_original

constrained

_problem.

Such a

terminology

is

actually

used inthe literature of

the traditional nonlinear

_programming

_(NLP)

_{[7, 8].}

Exact

_{augmented Lagrangian}

functions wereintroduced

by

Di Pillo and

Grippo

in

[7]

and

_[8],

_respectively

for

_{equality‐constrained}

and for

_{inequality‐constrained}

NLP

_problems.

Further

_{investigations}

had been done in

_[4,

_{9, 10, 11,}

_19].

_Recalling

that NSDP

problems

extend NLP

_problems,

herewe

_give

the first_steptowards the

_augmented

_Lagrangian

method

for NSDP. The

_proposed

function is

_basically

the classical

augmented Lagrangian

function for NSDP

_{problems, given}

in

_{[6, 21],}

withan additionaltermthat_guaranteestheexactness

property. Such term

_requires

afunction that estimates the

Lagrange mutipliers

associated

to a

point.

This estimator was

originally given

in

[14],

and extended further to NSDP

in

_[15].

We will show that the

_proposed

function is in fact _exact, if the

_{nondegeneracy}

condition issatisfiedin thefeasibleset of the NSDP.

We

_finally

observe that a more

general

class of these exact

_augmented

_Lagrangian

functions for NSDP is

_given

in

_[13].

In

_fact,

our _paper is

just

an _{easy note} associated

to

_it,

soallthe

_proofs

of the results described herecanbeseenin this

_original

_manuscript.

This_{paper is}

_organized

asfollows. In Section

_2,

we recall basic notations and definitions.

Theexact

_{augmented Lagrangian}

function is constructed in Section

_3,

and the exactness

resultsare

_given

inSection4. We concludeinSection

_5,

withsomefinal remarks and future

works.

2

Preliminaries

Let usfirst _present the mainnotations. We use _{x_{i}} and

Z_{ij}

to denote the ith elementofa

vectorx\in \mathbb{R}^{r} and

_{(i, j)}

_entry

_(ith

rowand

jth

column)

ofamatrixZ\in \mathrm{S}^{s},

respectively.

We

alsouse thenotation

_[xi]í

₌₁ and

_{[Z_{ij}]_{i,j=1}^{s}}

todenote x and Z_,

respectively.

For a function

p:\mathbb{R}^{S}\rightarrow \mathbb{R}

, its

gradient

and Hessian at a

point

x\in \mathbb{R}^{S} are

given

by

\nabla p(x)\in \mathbb{R}^{S}

and

\nabla^{2}p(x)\in \mathbb{R}^{\mathrm{s}\times s}

,

respectively.

For

q:\mathrm{S}^{\ell}\rightarrow \mathbb{R},

\nabla q(Z)

denotes the matrix with

(i,j)

term

given

by

the

_partial

derivatives

_{\partial q(Z)/\partial Z_{ij}}

. If

$\psi$:\mathbb{R}^{S}\times \mathrm{S}^{\ell}\rightarrow \mathbb{R}

,thenits

gradient

at

(x, Z)\in

\mathbb{R}^{S}\times \mathrm{S}^{\ell}

with _respect to x and Y are denoted

by

\nabla_{x} $\psi$(x, \mathrm{Y})\in \mathbb{R}^{S}

and

\nabla_{Y} $\psi$(x, Y)\in \mathrm{S}^{p},

respectively. Similary,

the Hessianof

$\psi$

at

_{(x, Z)}

with_respecttoxis writtenas

\nabla_{xx}^{2} $\psi$(x, \mathrm{Y})

.

For_anylinear_operator

_{\mathcal{G}:\mathbb{R}^{S}\rightarrow \mathrm{S}^{\ell}}

defined

_by

_{\displaystyle \mathcal{G}v=\sum_{i=1}^{8}v_{i}\mathcal{G}_{i}}

with

_{\mathcal{G}_{i}\in \mathrm{S}^{p},}

i=1,..._,s,

and v\in \mathbb{R}^{s}, the

adjoint

operator \mathcal{G}^{*} is defined

by

\mathcal{G}^{*z=}(\{\mathcal{G}_{1}, Z\}, \ldots, \{\mathcal{G}_{s}, Z\rangle)^{\mathrm{T}}, Z\in \mathrm{S}^{l}.

\nabla \mathcal{G}(x):\mathbb{R}^{8}\rightarrow \mathrm{S}^{p}

and defined

_by

\displaystyle \nabla \mathcal{G}(x)v=\sum_{i=1}^{s}v_{i}\frac{\partial \mathcal{G}(x)}{\partial x_{i}}, v\in \mathbb{R}^{s},

where

_{\partial \mathcal{G}(x)/\partial x_{i}\in \mathrm{S}^{l}}

are the

_partial

derivative matrices.

One

_important

_operator that is _necessary when

_dealing

with NSDP

problems

is the

Jordan

_product

associatedto the_space \mathrm{S}^{m}. For any

Y,

Z\in \mathrm{S}^{m}_, it isdefined

by

Y\displaystyle \circ Z:=\frac{YZ+ZY}{2}.

Taking

Y\in \mathrm{S}^{m},we also denote

by \mathcal{L}_{Y}:\mathrm{S}^{m}\rightarrow \mathrm{S}^{m}

the linear operator

given

by

\mathcal{L}_{Y}(Z):=Y\circ Z.

Sinceweare

only considering

the_space\mathrm{S}^{m} of

symmetric matrices,

wehave

\mathcal{L}_{Y}(Z)=\mathcal{L}_{Z}(Y)

.

Now,

let thetraceof Z\in \mathrm{S}^{m} be

_given

_by

_{\mathrm{t}\mathrm{r}(Z)}

_{:=\displaystyle \sum_{i=1}^{s}Z_{ii}}

and define

_{\langle Y, Z\rangle :=\mathrm{t}\mathrm{r}(YZ)}

asthe inner

product

of

symmetric

matrices Y and Z.

Then,

define L:\mathbb{R}^{n}\times \mathrm{S}^{m}\rightarrow \mathbb{R}asthe

Lagrangian

function associatedto

_problem

_(NSDP),

that

_is,

L(x, $\Lambda$):=f(x)-\{G(x) , $\Lambda$\rangle.

The

_pair

_{(x, $\Lambda$)\in \mathbb{R}^{n}\times \mathrm{S}^{m}}

satisfies the Karush‐Kuhn‐Tucker

_(KKT)

conditions of

_prob‐

lem

_{(NSDP) (or,}

it isa KKT

pair)

if the

following

conditions hold:

\nabla_{x}L(x, $\Lambda$) = 0,

$\Lambda$\circ G(x) = 0,

G(x) \in \mathrm{S}_{+}^{m},

$\Lambda$ \in \mathrm{S}_{+}^{m},

where

_{\nabla_{x}L(x, $\Lambda$)}

denotes the

_gradient

of L with_respectto x,that

is,

\nabla_{x}L(x, $\Lambda$)=\nabla f(x)-\nabla G(x)^{*} $\Lambda$.

The above conditions are _necessary for

optimality

undera constraint

qualification.

More‐

over, itcanbe shown that the condition

$\Lambda$\circ G(x)=0

canbe

replaced by

\{ $\Lambda$, G(x)\}=0

or

$\Lambda$ G(x)=0

because

_{G(x)\in \mathrm{S}_{+}^{m}}

and

_{$\Lambda$\in \mathrm{S}_{+}^{m}}

hold

_[22,

Section

_2].

In this_paper, we will

replace

the

problem

(NSDP)

with the

following problem,

which

is

_just

anonlinear

programming:

\mathrm{m}\mathrm{i}\dot{\mathrm{m}}\mathrm{m}x, $\Lambda$

ize

_{$\Psi$_{c}(x, $\Lambda$)}

(1)

subject

to

_{(x, $\Lambda$)\in \mathbb{R}^{n}\times \mathrm{S}^{m},}

where

$\Psi$_{c}:\mathbb{R}^{n}\times \mathrm{S}^{m}\rightarrow \mathbb{R}

,and c>0isa

penalty

parameter. Observe that the above

problem

is

unconstrained,

with both x and $\Lambda$ asvariables. As

usual,

we _say that a

_point

x\in \mathbb{R}^{n}

is

_stationary

of

_{$\Psi$_{c}}

when

_{\nabla$\Psi$_{\mathrm{c}}(x)=}

O. We use

G_{\mathrm{N}\mathrm{L}\mathrm{P}}(c)

and

L_{\mathrm{N}\mathrm{L}\mathrm{P}}(c)

to denote the sets of

global

and local

_minimizers,

respectively,

of the above

_problem.

We also define

_{G_{\mathrm{N}\mathrm{S}\mathrm{D}\mathrm{P}}}

and

L_{\mathrm{N}\mathrm{S}\mathrm{D}\mathrm{P}}

asthesetof

_global

andlocal minimizers of

_problem

_(NSDP),

_{respectively. Using}

such

Definition 1. A

_function

_{$\Psi$_{c}:\mathbb{R}^{n}\times \mathrm{S}^{m}\rightarrow \mathbb{R}}

is called an exact

_{augmented Lagrangian}

function

associatedto

_(NSDP)

_if,

and

_{only if,}

there exists \hat{c}>0

_satisfying

the

_following:

(a)

For all c>\hat{c},

if

(\overline{x},\overline{ $\Lambda$})\in G_{NLP}(c)

, then

\overline{x}\in G_{NsDP}

and

\overline{ $\Lambda$}

is a

corresponding Lagrange

multiplier.

Conversely, if

\overline{x}\in G_{NSDP}

with

\overline{ $\Lambda$}

as a

corresponding Lagrange multiplier,

then

_{(\overline{x},\overline{ $\Lambda$})\in G_{NLP}(c)}

_for

all c>\hat{c}.

(b)

Forall c>\hat{c},

if

(\overline{x},\overline{ $\Lambda$})\in L_{NLP}(c)

, then

\overline{x}\in L_{NSDP}

and

\overline{ $\Lambda$}

is a

corresponding Lagrange

multiplier.

Basically,

theabove definition shows that

_{$\Psi$_{\mathrm{c}}}

isanexact

_{augmented Lagrangian}

function when thereare

_equivalence

between the

global

_minimizers,

andif all local solutions of

(1)

are

local solutions of

_(NSDP),

for

_penalty

_{parameters greater}thanathreshold value. Itmeans

that the

_original

constrained conic

_problem

_(NSDP)

canbe

replaced

withanuncontrained

nonlinear

_{programming problem}

₍₁₎

when the

_penalty

_parameter is chosen

_{appropriately.}

In orderto construct suchanexact

_{augmented Lagrangian function,}

wewill_suppose that

the

_following

_assumption

holdsinthe whole_paper. Itiswell‐known that the

_{nondegeneracy}

condition,

defined

_below,

extends the classical linear

independence

constraint

_{qualification}

for nonlinear

_programming

_{[5, 20].}

Assumption

2.

_Every

x\in \mathbb{R}^{n}

feasible

for

(NSDP)

\dot{u}

_{nondegenerate,}

that _is,

\mathrm{S}^{m}=1\mathrm{i}\mathrm{n}T_{\mathrm{S}_{+}^{m}}(G(x))+{\rm Im}\nabla G(x)

,

where

_{T_{\mathrm{S}_{+}^{m}}(G(x))}

denotes the _tangent cone

of

\mathrm{S}_{+}^{m}

at

_G(x)

,

{\rm Im}\nabla G(x)

is the

image

of

the linearmap

\nabla G(x)

, and lin means

lineality

space.

3

The

_proposed

_{augmented Lagrangian}

function

Theexact

_augmented

_Lagrangian

function consideredin

_[8]

takes intoaccountanestimation

of the

_{Lagrange multipliers, given}

in

_[14].

An extension ofit for NSDP

_problems

were

proposed

in

_[15].

Given x\in \mathbb{R}^{n},wesolve the

following

unconstrained

problem:

\displaystyle \min_{ $\Lambda$}

imize

_{\Vert\nabla_{x}L(x, $\Lambda$)\Vert^{2}+$\zeta$_{1}^{2}\Vert \mathcal{L}_{G(x)}( $\Lambda$)\Vert_{F}^{2}+$\zeta$_{2}^{2}r(x)\Vert $\Lambda$\Vert_{F}^{2}}

(2)

subject

to

_{$\Lambda$\in \mathrm{S}^{m},}

where

_{$\zeta$_{1},}

_{$\zeta$_{2}\in \mathbb{R}}

are

_{positive scalars,}

\Vert\cdot\Vert

denotes the Euclidean_norm,

\Vert\cdot\Vert_{F}

isthe Frobenius

norm, and r:\mathbb{R}^{n}\rightarrow \mathbb{R} denotes the residual function associatedtothe feasibleset, that

_is,

r(x) :=\displaystyle \frac{1}{2}\Vert P_{\mathrm{S}_{+}^{m}}(-G(x))\Vert_{F}^{2}=\frac{1}{2}\Vert P_{\mathrm{S}_{+}^{m}}(G(x))-G(x)\Vert_{F}^{2},

with

_{P_{\mathrm{S}_{+}^{m}}}

_denoting

the

_projection

onto

_{\mathrm{S}_{+}^{m}}

. Observe that

r(x)=0

if,

and

only if,

x is

feasible for

_(NSDP).

Lemma 3.

_[15,

Lemma 2.2 and

_Proposition

_2.31

_Suppose

that

_Assumption

2 holds. For

anyx\in \mathbb{R}^{n},

define

N:\mathbb{R}^{n}\rightarrow \mathrm{S}^{m} as

N(x) :=\nabla G(x)\nabla G(x)^{*}+$\zeta$_{1}^{2}\mathcal{L}_{G(x)}^{2}+$\zeta$_{2}^{2}r(x)I

,

(3)

whereI denotes the

_identity

matrix.

_Then,

the

_following

statementsare true.

(a)

N is

_{continuosly differentiable}

and

_for

all x\in \mathbb{R}^{n}, the matrix

N(x)

is

positive

definite.

(b)

The solution

_{of problem}

₍₂₎

is

_unique

andit is

_given

_by

$\Lambda$(x)=N(x)^{-1}\nabla G(x)\nabla f(x)

.

(4)

(c)

If

(x, $\Lambda$)\in \mathbb{R}^{n}\times \mathrm{S}^{m}

is a KKT

_pair

of

(NSDP),

then

$\Lambda$(x)= $\Lambda$.

(d)

The _operator $\Lambda$ is

_{continuously differentiable,}

and

_{\nabla $\Lambda$(x)=N(x)^{-1}Q(x)}

, where

Q(x) := \nabla^{2}G(x)\nabla_{x}L(x, $\Lambda$(x))+\nabla G(x)\nabla_{xx}^{2}L(x, $\Lambda$(x))

-$\zeta$_{1}^{2}\nabla_{x}[\mathcal{L}_{G(x)}^{2}( $\Lambda$(x))]-$\zeta$_{2}^{2}\nabla r(x) $\Lambda$(x)

.

Basedon the above

_lemma,

we _proposethe

_following

_{augmented Lagrangian}

function.

L_{\mathrm{c}}(x, $\Lambda$) :=f(x)+\displaystyle \frac{1}{2c}(\Vert P_{\mathrm{S}_{+}^{m}}( $\Lambda$-cG(x))\Vert_{F}^{2}-\Vert $\Lambda$\Vert_{F}^{2})+\Vert N(x)( $\Lambda$(x)- $\Lambda$)\Vert_{F}^{2}

.

(5)

Note thatit is

_equivalent

tothe

_augmented

_Lagragian

function for nonlinear semidefinite

programs, except for the lastterm

\Vert N(x)( $\Lambda$(x)- $\Lambda$)\Vert_{F}^{2}

. From

(3)

and

(4),

observe that

N(x)( $\Lambda$(x)- $\Lambda$)

=

\nabla G(x)\nabla f(x)-\nabla G(x)\nabla G(x)^{*} $\Lambda-\zeta$_{1}^{2}\mathcal{L}_{G(x)}^{2}( $\Lambda$)-$\zeta$_{2}^{2}r(x) $\Lambda$

= \nabla G(x)\nabla_{x}L(x, $\Lambda$)-$\zeta$_{1}^{2}\mathcal{L}_{G(x)}^{2}( $\Lambda$)-$\zeta$_{2}^{2}r(x) $\Lambda$

.

(6)

Also,

consider the

_following

auxiliar function

Y_{c}:\mathbb{R}^{n}\times \mathrm{S}^{m}\rightarrow \mathrm{S}^{m}

defined

_by

Y_{c}(x, $\Lambda$):=P_{\mathrm{S}_{+}^{m}}(\displaystyle \frac{ $\Lambda$}{c}-G(x))-\frac{ $\Lambda$}{c}.

Then,

the

_gradient

of

_{L_{c}(x, $\Lambda$)}

with _respectto xis

_{given by}

\nabla_{x}L_{c}(x, $\Lambda$)

=

\nabla f(x)-\nabla G(x)^{*}P_{\mathrm{S}_{+}^{m}}( $\Lambda$-cG(x))+2K(x, $\Lambda$)^{*}N(x)( $\Lambda$(x)- $\Lambda$)

= \nabla_{x}L(x, $\Lambda$)-c\nabla G(x)^{*}Y_{c}(x, $\Lambda$)+2K(x, $\Lambda$)^{*}N(x)( $\Lambda$(x)- $\Lambda$)

, with

K(x, $\Lambda$) := \nabla_{x}[N(x)( $\Lambda$(x)- $\Lambda$)]

= \nabla_{x}[\nabla G(x)\nabla_{x}L(x, $\Lambda$)-$\zeta$_{1}^{2}\mathcal{L}_{G(x)}^{2}( $\Lambda$)-$\zeta$_{2}^{2}r(x) $\Lambda$],

where the second

_equality

follows from

_(6).

Some calculations show that

\nabla_{x}L_{c}(x, $\Lambda$)=\nabla_{x}L(x, $\Lambda$)-c\nabla G(x)^{*}Y_{c}(x, $\Lambda$)+2\nabla_{xx}^{2}L(x, $\Lambda$)\nabla G(x)^{*}N(x)( $\Lambda$(x)- $\Lambda$)

+2[\displaystyle \langle\frac{\partial^{2}G(x)}{\partial x_{i}x_{j}}, N(x)( $\Lambda$(x)- $\Lambda$)\rangle]_{i,j=1}^{n}\nabla_{x}L(x, $\Lambda$)

-2$\zeta$_{1}^{2}[\displaystyle \langle\frac{\partial G(x)}{\partial x_{i}}\circ(G(x)\circ $\Lambda$)+G(x)\circ(\frac{\partial G(x)}{\partial x_{i}}\circ $\Lambda$) , N(x)( $\Lambda$(x)- $\Lambda$ i=1n

-2$\zeta$_{2}^{2}\langle $\Lambda$, N(x)( $\Lambda$(x)- $\Lambda$)\rangle\nabla r(x)

.

Moreover,

the

_gradient

of

_{L_{c}(x, $\Lambda$)}

with _respectto Acanbe writtenasfollows:

4

Exactness results

In this

_section,

wewill first establish the relation between the KKT

points

of the

original

(NSDP)

problem

with the

_following

unconstrainedone:

\displaystyle \min_{x}

imize

_{L_{c}(x, $\Lambda$)}

(7)

subject

to

_{(x, $\Lambda$)\in \mathbb{R}^{n}\times \mathrm{S}^{m}.}

We referto

_[13]

for details and

_proofs

of the

_subsequent

results. As itcanbeseeninthenext

three

_{propositions,}

aKKT

pair

of

(NSDP)

is

stationary

of

(7),

but theconverse

implication

only

holds when the

_penalty

_parameteris

greater

than a thresholdvalue.

Moreover,

asin

the classical

augmented Lagrangian

methodsorexact

_penalty

methods

_[1],

we_mayalso end

upwitha

_{stationary point}

of the residual functionrthat is infeasible for

(NSDP).

Proposition

4.

_If

_{(x, $\Lambda$)\in \mathbb{R}^{n}\times \mathrm{S}^{m}}

is a KKT

_pair

_of

_(NSDP),

_then,

_for

all c>0, it is

stationaw

of

(7),

that

_is,

_{\nabla_{x}L_{c}(x, $\Lambda$)=0}

and

_{\nabla_{ $\Lambda$}L_{\mathrm{c}}(x, $\Lambda$)=0.}

Proposition

5. Let\hat{x}\in \mathbb{R}^{n} be

_{feasible for}

_(NSDP).

Assume that there exist

_{\{x^{k}\}\subset \mathbb{R}^{n},}

\{$\Lambda$_{k}\}\subset \mathrm{S}^{m}

, and

\{ck\}\subset \mathbb{R}++

with

x_{k}\rightarrow\hat{x}

and c_{k}\rightarrow\infty such that

(x_{k}, $\Lambda$_{k})

is

stationary

of

(7)

for

all k.

Then,

there is

\hat{k}>0

such that

(x^{k}, $\Lambda$_{k})

is KKT

of

(NSDP)

for

all

k>\hat{k}.

Proposition

6. Let

_{\{x^{k}\}\subset \mathbb{R}^{n},}

_{\{$\Lambda$_{k}\}\subset \mathrm{S}^{m}}

, and

\{ck\}\subset \mathbb{R}++be

sequences such that

c_{k}\rightarrow\infty and

_{(x $\Lambda$)}

is

_{stationary of}

₍₇₎

_for

all k. Assume that there is a

subsequence

\{x^{k_{j}}\}

of

\{x^{k}\}

such that

x^{k_{j}}\rightarrow\hat{x}

_for

some \hat{x}\in \mathbb{R}^{n}.

Then,

either there exists

\hat{k}>0

such

that

_{(x^{k_{j}}, $\Lambda$_{k_{j}})}

is a KKT

pair

of

(NSDP)

for

all

k_{j}>\hat{k}

, or\hat{x} is a

stationary point of

the residual

_function

r that is

infeasible

for

(NSDP).

We now use the notations

G_{\mathrm{N}\mathrm{S}\mathrm{D}\mathrm{P}}(L_{\mathrm{N}\mathrm{S}\mathrm{D}\mathrm{P}})

and

G_{\mathrm{N}\mathrm{L}\mathrm{P}}(c)(L_{\mathrm{N}\mathrm{L}\mathrm{P}}(c))

for the sets of

_global

(local)

minimizersof

problems

(NSDP)

and

_(7),

_{respectively.}

The

_following

theorems show

thatthe

_proposed

function

L_{\mathrm{c}} given

in

₍₅₎

isin factan exact

_augmented

_Lagrangian

func‐

tion.

Theorem 7. Let

_{\{x^{k}\}\subset \mathbb{R}^{n},}

_{\{$\Lambda$_{k}\}\subset \mathrm{S}^{m}}

, and

\{ck\}\subset \mathbb{R}++be

sequencessuch that c_{k}\rightarrow\infty and

_{(x_{k},$\Lambda$_{k})\in L_{NLP}(c_{k})}

_for

all k. Assume that thereis a

subsequence

\{x^{k_{\mathrm{j}}}\}

of

\{x^{k}\}

such

that

_{x^{k_{j}}\rightarrow\hat{x} for}

some \hat{x}\in \mathbb{R}^{n}.

Then,

either there exists

\hat{k}>0

such that

x^{k_{j}}\in L_{NSDP},

withan associated

Lagrange

multiplier

_{$\Lambda$_{k_{j}}}

for

all

k_{j}>\hat{k}

, or\hat{x} is a

stationary point

of

the residual

_function

r that is

infeasible for

(NSDP).

Theorem 8. Assume that there exists \overline{c}>0 such that

_{\displaystyle \bigcup_{\mathrm{c}\geq\overline{\mathrm{c}}}G_{NLP}(c)}

is bounded.

_Then,

there exists\hat{c}>0 such that

_{G_{NLP}(c)=G_{NSDP}}

_for

allc\geq\hat{c}.

5

Final remarks

We

_proposed

an exact

_augmented

_Lagrangian

function for

_general

nonlinear semidefinite

programming problems,

and we

proved

theexactnessresult under the

_{nondegeneracy}

con‐

dition.

_Thus,

_by

_solving

the unconstrained

_problem

₍₇₎

withan

_appropriate

penalty

_param‐

can be solved

_using

asecond‐order

method,

like the semismooth Newton. Insuch a_case,

convergenceresults should be

established,

and a_{way to} avoid the third‐ordertermsof the

problem data,

that appear inthe Hessianof

L_{\mathrm{c}}

,should be also considered. These facts and

the numerical

_experiments

are under

investigation.

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