Duality for Multiobjective Programming Problems with Equilibrium
Constraints on Hadamard Manifolds under Generalized Geodesic
Convexity
BALENDU BHOOSHAN UPADHYAY1, ARNAV GHOSH1, I. M. STANCU-MINASIAN2
1Department of Mathematics,
Indian Institute of Technology Patna,
INDIA
2‘‘Gheorghe Mihoc-Caius Iacob”,
Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy,
Bucharest,
ROMANIA
Abstract: - This article is devoted to the study of a class of multiobjective mathematical programming problems
with equilibrium constraints on Hadamard manifolds (in short, (MPPEC)). We consider (MPPEC) as our primal
problem and formulate two different kinds of dual models, namely, Wolfe and Mond-Weir type dual models
related to (MPPEC). Further, we deduce the weak, strong as well as strict converse duality relations that relate
(MPPEC) and the corresponding dual problems employing geodesic pseudoconvexity and geodesic
quasiconvexity restrictions. Several suitable numerical examples are incorporated to demonstrate the
significance of the deduced results. The results derived in this article generalize and extend several previously
existing results in the literature.
Key-Words: - Multiobjective optimization, Guignard constraint qualification, Generalized geodesic convexity,
Duality, Hadamard manifolds
Received: August 11, 2022. Revised: February 27, 2023. Accepted: March 23, 2023. Published: April 26, 2023.
1 Introduction
In recent times, the study of optimization problems
in the setting of manifolds has emerged as a very
significant area of research. It is possible to model
various practical problems that arise in numerous
areas related to engineering in a much more
effective manner on the setting of a manifold, rather
than that of Euclidean space, see, [1], [8]. In fact,
extending and generalizing the methods of
optimization from Euclidean spaces to manifolds
have various important advantages from theoretical
as well as practical standpoints. For instance,
numerous non-convex mathematical programming
problems can be converted into convex
mathematical programming problems by employing
the Riemannian geometry perspective (see, for
instance, [17], [18]). Furthermore, it can be
observed that the relative interior of several
important constraints in certain mathematical
programming problems can be viewed as Hadamard
manifolds, for instance, the positive orthant
(see, for instance, [15]) equipped with the Dikin
metric
, and the hypercube
(see, for instance, [16]) equipped with the
metric =diag (
are Hadamard manifolds.
As a result, several constrained optimization
problems can be suitably transformed into much
simpler unconstrained problems by appropriate use
of the Riemannian geometry. Due to this fact, a
wider range of optimization problems can be
investigated by formulating the problems in the
framework of Riemannian and Hadamard
manifolds. The notions of geodesic convex sets and
geodesic convex function in manifold setting are
developed to generalize the definitions of convex
sets and convex functions (see, [21], [26]). Further,
the notions of geodesic pseudoconvex and geodesic
quasiconvex functions were introduced in [26], by
generalizing geodesic convex functions in the
setting of Riemannian manifolds. In the last few
years, several authors have extended many
interesting ideas of mathematical programming
from Euclidean spaces to the setting of Riemannian
as well as Hadamard manifolds, see, [3], [5], [13],
[29], [30], [31], [32], and the references cited
therein.
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In the theory of mathematical programming, an
optimization problem that is accompanied by some
complementarity constraints, or certain variational
inequality constraints is termed a mathematical
programming problem with equilibrium constraints
(in brief, (MPEC)). One of the first attempts in
investigating such optimization problems is due to
[7], where the existence of efficient solutions for
(MPECs) is explored. Due to its immense scope of
applicability in numerous fields of science,
technology, and engineering (see, for instance, [19],
[20]), (MPECs) have been studied by numerous
authors in recent years. For further details and an
updated survey of (MPEC) and its applications, we
refer the readers to [12], [14], [22], [23], [24], [27],
[28], and the references cited therein.
Several regularity and optimality conditions for
(MPECs) were deduced in [4]. The existence of
efficient solutions for (MPECs) was investigated in
[7]. In [6], the Wolfe type duality model for
(MPECs) was explored and several interesting
duality results were derived. Optimality conditions
and duality for semi-infinite (MPECs) were studied
in [12]. Several duality results for multiobjective
(MPECs) were derived in [23]. Further, optimality
criteria and duality for multiobjective (MPECs)
were studied in [24]. Recently, optimality criteria
for multiobjective (MPEC) on Hadamard manifolds
were derived in [25].
Motivated by the results derived in [6], [14],
[23], [24], in this article we consider a certain class
of multiobjective mathematical programming
problem with equilibrium constraints on the
framework of Hadamard manifolds (in short,
((MPPEC)) as our primal problem. We formulate
two different kinds of dual problems related to
(MPPEC), namely, Wolfe and Mond-Weir type dual
problems. Further, we deduce weak, strong as well
as strict converse duality relations that relate
(MPPEC) and the corresponding dual problems
under geodesic quasiconvexity and pseudoconvexity
restrictions. To the best of our knowledge, this is for
the first time that duality results for (MPPEC) have
been investigated in the context of Hadamard
manifolds.
The main contributions and novelty of the work
in this article are twofold. Firstly, the results in this
paper generalize the corresponding duality results
deduced in [24], in the setting of a more general
space, that is, Hadamard manifolds. The results
obtained in this article extend the corresponding
results of [23], from Euclidean space to the context
of Hadamard manifolds. Secondly, the duality
results obtained in this article also extend the
corresponding duality results derived in [6], for a
more general category of optimization problems,
that is, (MPPEC) and generalize them on the
framework of a wider space, which is Hadamard
manifolds.
This article is organized in the following
manner. Some basic mathematical preliminaries and
concepts are recalled in Section 2. Moreover, we
discuss the generalized Guignard constraint
qualification and Karush-Kuhn-Tucker type
necessary optimality criteria for (MPPEC). In
Section 3, we formulate the Wolfe type dual
problem related to (MPPEC) and deduce the weak,
strong as well as strict converse duality relations
that relate (MPPEC) and the dual problem
employing geodesic pseudoconvexity restrictions. In
Section 4, the Mond-Weir dual problem related to
(MPPEC) is formulated, and weak, strong as well as
strict converse duality relations that relate (MPPEC)
and the dual problem are derived using geodesic
pseudoconvexity and quasiconvexity assumptions.
We conclude our discussions in Section 5 along
with some future research directions.
2 Problem Formulation
In this section, we recollect some notation,
preliminary definitions, and concepts that will be
used in the rest of the paper.
The dimensional Euclidean plane is
indicated by using the standard symbol . The set
containing every natural number is signified by .
The symbol is used to denote any empty set. The
symbol is employed to indicate the standard
Euclidean inner product on . Let be
arbitrary pair of vectors in . The following
notation for inequalities will be employed in this
article:
and
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We now recollect some basic definitions and
concepts from Riemannian manifolds as well as
Hadamard manifolds which will be required in the
sequel.
We shall be using the notation to signify a
smooth manifold having dimension where is
any natural number. Let be arbitrary. The set
that contains every tangent vector at the element
is known as the tangent space at , and is
denoted by . For any element , is
a real vector space, having a dimension In case
we are restricted to real manifolds, is
isomorphic to the dimensional Euclidean space
.
A Riemannian metric, denoted by on the set
is a 2-tensor field that is symmetric as well as
positive-definite. For every pair of elements
, the inner product of and is
given by:
where the symbol denotes the Riemannian metric
at the element . The norm corresponding to
the inner product is denoted by (or
simply , when there is no ambiguity regarding
the subscript).
Let and be any
piecewise differentiable curve that joins the
elements and in , that is, we have:
The length of the curve is denoted by and is
defined in the following manner:
For any differentiable curve , a vector field is
referred to be parallel along the curve , provided
that the following condition is satisfied:
If , then is termed as a geodesic. If
, then the curve is said to be normalised.
For any , the exponential function
is given by ,
where is a geodesic that satisfies and
. A Riemannian manifold is referred
to as geodesic complete, provided that the
exponential function is defined for every
arbitrary
and .
A Riemannian manifold is referred to as a
Hadamard manifold (or, Cartan-Hadamard
manifold) provided that is simply connected,
geodesic complete, as well as has a nonpositive
sectional curvature throughout. Henceforth, in our
discussions, the notation will always signify a
Hadamard manifold of dimension , unless it is
specified otherwise.
Let be some arbitrary element lying in
the Hadamard manifold Then, the exponential
function on the tangent space
is a
globally diffeomorphic function. Moreover, the
inverse of the exponential function
satisfies
. Furthermore, for
every pair of arbitrary elements
there
will always exist some unique normalized minimal
geodesic
such that the geodesic
satisfies the following:
Thus, every Hadamard manifold of
dimension is diffeomorphic to the -dimensional
Euclidean space . The gradient of any smooth
function is symbolized by grad and is
a vector field on that is defined as:
where is also some vector field on the manifold
.
The following definition is from [26].
Definition 2.1. Any non-empty subset of a
Hadamard manifold is termed as a geodesic
convex set in , if for every and for every
geodesic joining the points and
we have
where,
The following definitions are from [2].
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Definition 2.2. Let be a geodesic convex
set. Let be any real valued function on
the set
(i) The function is termed as a geodesic
(respectively, strictly geodesic) pseudoconvex
function at provided that for any arbitrary
element (respectively,), we have
(ii) The function is termed as a geodesic
quasiconvex function at , provided that for
any arbitrary element , we have
Remark 1. (i) If , then
, where denotes the gradient of the
function at in , and . In this
case, the definitions of geodesic pseudoconvex and
quasiconvex functions correspond to the usual
standard definitions of differentiable pseudoconvex
and quasiconvex functions (see, for instance, [10],
pp. 146) for Euclidean spaces.
(ii) If the function is a geodesic convex
function, then the function is automatically
geodesic pseudoconvex as well as geodesic
quasiconvex (see, Definition 10.1 in [26], and
Definition 13.2.1 in [21]).
For further detailed exposition on geodesic
quasiconvexity and pseudoconvexity in the setting
of Hadamard manifolds, we refer to [21], [26].
Unless specified otherwise, we shall employ the
notation to denote an dimensional Hadamard
manifold.
In this article, the following multiobjective
mathematical programming problem with
equilibrium constraints on the Hadamard manifold
is considered:
subject to ,
,
,
,
,
where every component of the objective function
, and constraints
are real-valued and smooth functions
defined on some dimensional Hadamard
manifold , where is a natural number. We use
the symbol to indicate the set of all feasible
solutions to (MPPEC).
Let be any arbitrary feasible element. We
now define a few index sets as follows that will
render the subsequent analysis convenient:
The following definition will be employed in the
sequel.
Definition 2.3. ([11]). Any arbitrary feasible
element is termed as a Pareto efficient (resp.,
weak Pareto efficient) solution of (MPPEC),
provided that there does not exist any other feasible
element that satisfies the following:
For any arbitrary feasible element , we now
define the sets (for every ) and that will
be used in the discussion that follows.
Remark 2. (i) From the above definitions of the sets
and , it is clear that
(ii) In case then (MPPEC) reduces to a
single-objective (MPEC). In such cases, we have
The following definition of the Bouligand tangent
cone on the Hadamard manifold is from [9].
Definition 2.4. ([9]). Let and
The contingent cone (in other terms, Bouligand
tangent cone) of at is symbolized by the
notation and is the set defined as follows:
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The following definitions and theorem are from
[25].
Definition 2.5. Let be any arbitrary feasible
element. The modified linearizing cone to the set
at the feasible element is the set defined as
follows:
Definition 2.6. Let us assume that is any
arbitrary feasible element of (MPPEC). The
generalized Guignard constraint qualification (in
short, (GGCQ)) is said to hold at provided that the
following inclusion relation is satisfied:
Theorem 2.7. Let be a Pareto efficient
solution of (MPPEC). Moreover, let us suppose that
(GGCQ) holds at . Then there exist real numbers
which satisfies the
following:
(1)
(2)
3 Wolfe Duality
In this section, the Wolfe type dual problem related
to (MPPEC) is formulated. Subsequently, we prove
the weak, strong as well as strict converse duality
relations that relate (MPPEC) and the dual problem
employing certain geodesic pseudoconvexity
assumptions.
Let us now consider that
and
. The Wolfe type dual model (in
brief, (WDP)), related to (MPPEC) may be
formulated as:
(WDP) Maximize
subject to
, (3)
(4)
The set of all feasible elements of (WDP) is denoted
by We define an auxiliary function
as follows:
for every
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In the following theorem, we establish weak duality
relation that relates (MPPEC) and (WDP).
Theorem 3.1. Let and . Let
be geodesic pseudoconvex at . Then the inequality
does not hold.
Proof. On contrary, we suppose that
. Then there exists some such that
for all and the above inequality holds
strictly for . Since , ,
it follows from the above inequality
that
From the definition of , it follows that
By invoking the geodesic pseudoconvexity
restriction on at , we get
which is a contradiction to (3). Thus, the proof is
complete.
In the following theorem, we establish strong
duality relation that relates (MPPEC) and (WDP).
Theorem 3.2. Let be any Pareto efficient
solution of (MPPEC). Let us further suppose that
(GGCQ) holds at . Then there exist some
, such
that
and Further, if every assumption
of the weak duality theorem (Theorem 3.1) holds,
then is a Pareto efficient solution of (WDP).
Proof. Since (GGCQ) is satisfied with the Pareto
efficient solution , it follows from Theorem
2.7 that there exist some
such that equations (1)
and (2) of Theorem 2.7 are satisfied. From the
feasibility conditions of (MPPEC), it follows that
This shows that and
On contrary, suppose that is not
a Pareto efficient solution of (WDP). Then there
exists , such that
which contradicts
the weak duality theorem (Theorem 3.1). Thus, the
proof is complete.
In the following theorem, we establish the strict
converse duality relation that relates (MPPEC) and
(WDP).
Theorem 3.3. Let and be
arbitrary feasible elements of (MPPEC) and (WDP),
respectively. Let us assume that the following
inequality holds:
If the assumption of weak duality theorem
(Theorem 3.1) is satisfied, then
Proof. On contrary, let Given that
From the feasibility conditions and definitions of
index sets, we infer that
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By invoking the geodesic pseudoconvexity
restriction on at , we get
which is a contradiction to (3). Thus, the proof is
complete.
Remark 3. (i) If , then Theorem 3.1 and
Theorem 3.2 reduce to Theorem 4 and Theorem 5
derived in [24].
(ii) The weak, strong as well as strict converse
duality relations (Theorem 3.1, Theorem 3.2 and
Theorem 3.3) extends Theorem 3, Theorem 4 and
Theorem 5 of [23], on wider space, that is,
Hadamard manifold, and generalize it for
(MPPEC).
In the following numerical example, we
demonstrate the results of Mond-Weir duality on the
Poincaré half plane, which is a Hadamard manifold
with negative sectional curvature.
Example 3.4. Consider the Poincaré half plane,
which is the set defined as
is then a Riemannian manifold (see,
for instance, [26]). The tangent space at every
element is given by. The
Riemannian metric on the set is given by
where
.
Furthermore, it can also be verified that is also a
Hadamard manifold having a sectional curvature of
Consider the following problem (P) on the set
which is a (MPPEC):
(P) Minimize
subject to
where the functions
are considered to be smooth functions on . The
set of feasible elements for (P) is
The Wolfe type dual problem related to (P) may be
formulated in the following manner:
(WD) Maximize
subject to
where,
The set containing all feasible elements of (WD) is
denoted by Consider the element
. By simple calculations, = is a Pareto
efficient solution of the problem (P). Let us now
choose .
We observe that the constraint
is satisfied. Hence, Furthermore, we
have It can be verified that
is geodesic pseudoconvex at . Thus, it is
illustrated that every assumption and implication of
the weak duality theorem is verified.
4 Mond-Weir Duality
In this section, the Mond-Weir type dual problem
related to (MPPEC) is formulated. Subsequently, we
deduce the weak, strong, as well as strict converse
duality relations that relate (MPPEC) and the dual
problem employing certain generalized geodesic
quasiconvexity and pseudoconvexity assumptions.
Let The Mond-Weir type dual model (in
brief, (MWD)) related to (MPPEC) may be
formulated as:
(MWD) Maximize
subject to
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(4)
and
where
The set containing every feasible element of
(MWD) is signified by the symbol
The following index sets will be helpful in deriving
duality results in the rest of the paper.
Now, we derive weak duality relations that relate
(MPPEC) and (MWD).
Theorem 4.1. Let and Let us
suppose that the functions
are geodesic quasiconvex at . Further, let
and
be strictly
geodesic pseudoconvex at . Then the inequality
does not hold true.
Proof. On contrary, let us suppose that
. Then, as , it follows that
From the geodesic strict pseudoconvexity of
, it follows that
(5)
For every , we have
Then, in light of the geodesic quasiconvexity
assumption on , we obtain the following:
For every , we have
Then, in view of the geodesic quasiconvexity
assumption on , and definition of index sets, we
obtain
Again, and
From the
geodesic quasiconvexity assumption on and
and definitions of index sets, we obtain
Since by hypothesis , it
follows from above inequalities that
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By combining each of the inequalities obtained
above, we get the following expression:
. (6)
It follows from (4) and (6) that
, which is a
contradiction to (5). Thus, the proof is complete.
Now, we establish strong duality relation that relates
(MPPEC) and (MWD).
Theorem 4.2. Let be a Pareto efficient
solution of (MPPEC). Let us further suppose that
(GGCQ) is satisfied at . Then there exist some
, such
that and the corresponding objective
function values are equal. Moreover, if every
assumption of weak duality (Theorem 4.1) is
satisfied, then is a Pareto efficient solution
of (MWD).
Proof. Since (GGCQ) is satisfied at the Pareto
efficient solution , it follows from Theorem
2.7 that there exist some
, such that equations
(1) and (2) of Theorem 2.7 are satisfied. From the
feasibility conditions of (MPPEC) it follows that
and the corresponding objective
function values are equal.
On contrary, let us suppose that is not a
Pareto efficient solution of (MWD). Then there
exists , such that
which is a contradiction to the weak duality theorem
(Theorem 4.1). Thus, the proof is complete.
Now, we deduce strict converse duality relation that
relates (MPPEC) and (MWD).
Theorem 4.3. Let and be
arbitrary feasible elements of (MPPEC) and
(MWD), respectively. Let us assume that the
following inequality holds:
If the assumption of weak duality theorem
(Theorem 4.1) is satisfied, then
Proof. On contrary, let Given that
By invoking the geodesic strict pseudoconvexity of
it follows that
For every , we have
Then, in light of the geodesic quasiconvexity
assumption on we obtain the following
For every , we have Then,
in view of the geodesic quasiconvexity assumption
on , and definition of index sets, we obtain
Again, and
From the
geodesic quasiconvexity assumption on and
and definitions of index sets, we obtain
Since by hypothesis, it
follows from above inequalities that
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By combining each of the inequalities obtained
above, we obtain the following expression:
(7)
It follows from (4) and (7) that
, which is a
contradiction. Thus, the proof is complete.
Remark 4. (i) If , then Theorem 4.1 and
Theorem 4.2 reduce to Theorem 6 and Theorem 7
derived in [24], for Euclidean spaces.
(ii) The weak, strong as well as strict converse
duality relations (Theorem 4.1, Theorem 4.2 and
Theorem 4.3) extend Theorem 6, Theorem 7
and Theorem 8, respectively, derived in [23], on the
framework of wider space, namely, Hadamard
manifold, and generalize it for (MPPEC).
In the following numerical example, we illustrate
the results derived for Mond-Weir duality.
Example 4.4. Consider the problem (P) as
formulated in Example 3.4.
The Mond-Weir dual problem related to (P),
denoted by (MWD), may be formulated as follows
(MWD) Maximize
subject to
where
The feasible set of (MWD) is
denoted by Consider the element
. By simple calculations, it can be
verified that is a Pareto efficient solution of the
problem (P). Then by choosing multipliers as
, we observe
that Moreover, the functions
are geodesic quasiconvex and
is strictly geodesic pseudoconvex at .
Thus, it can be verified that every assumption and
implication of the weak duality theorem is valid.
5 Conclusion
In this article, we have investigated a certain
category of multiobjective mathematical
programming problems with equilibrium constraints
on Hadamard manifolds (abbreviated as, (MPPEC)).
We have formulated the Wolfe type dual model
(WDP) and Mond-Weir type dual model (MWD)
related to (MPPEC) and derived the weak, strong, as
well as strict converse duality relations that relate
(MPPEC) and the dual models under generalized
geodesic convexity restrictions. Several non-trivial
numerical examples have been provided to
demonstrate the importance of the derived results.
The various results derived in this article extend
and generalize several notable results present in the
literature. For instance, the results that are
established in this article generalize the
corresponding results deduced in [24], on the
framework of an even wider space, which is,
Hadamard manifolds, as well as for an even more
class of convex functions. Further, the results
deduced in this article extend the duality results
deduced in [23], to Hadamard manifolds. Moreover,
the results established in the paper also extend the
results derived in [6], to a more general class of
programming problems, namely, (MPPEC), and
generalize them in the context of a wider space,
namely, Hadamard manifolds.
For future work, we would like to extend the
duality results derived in this article for nonsmooth
optimization problems with equilibrium constraints
in the setting of Hadamard manifolds. Moreover, it
would be an exciting challenge to study duality
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results for mathematical programming problems
with vanishing constraints on Hadamard manifolds.
Acknowledgments:
The authors are thankful to the anonymous referees
for their valuable comments and suggestions which
helped to improve the presentation of the paper.
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Arnav Ghosh, I. M. Stancu-Minasian
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
-Balendu Bhooshan Upadhyay is responsible for the
conceptualization of the research problem as well as
the supervision of the work.
-Arnav Ghosh is responsible for formal analysis and
writing the first draft of the paper.
-I. M. Stancu-Minasian revised and edited the first
draft of the paper.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The second author is supported by the Council of
Scientific and Industrial Research (CSIR), New
Delhi, India, through grant number
09/1023(0044)/2021-EMR-I.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.31
Balendu Bhooshan Upadhyay,
Arnav Ghosh, I. M. Stancu-Minasian
E-ISSN: 2224-2880
270
Volume 22, 2023
