{'fraction_non_alphanumeric': 0.051766237994465246, 'fraction_numerical': 0.008302132508546313, 'mean_word_length': 4.324593716143012, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 7, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
GLOBAL OPTIMALITY CONDITIONS AND DUALITY THEOREMS FOR ROBUST OPTIMAL SOLUTIONS OF OPTIMIZATION PROBLEMS WITH DATA UNCERTAINTY, USING UNDERESTIMATORS

. In this paper, a robust optimization problem, which features a maximum function of continuously diﬀerentiable functions as its objective func- tion, is investigated. Some new conditions for a robust KKT point, which is a robust feasible solution that satisﬁes the robust KKT condition, to be a global robust optimal solution of the uncertain optimization problem, which may have many local robust optimal solutions that are not global, are established. The obtained conditions make use of underestimators, which were ﬁrst introduced by Jayakumar and Srisatkunarajah [1, 2] of the Lagrangian associated with the problem at the robust KKT point. Furthermore, we also investigate the Wolfe type robust duality between the smooth uncertain optimization problem and its uncertain dual problem by proving the suﬃcient conditions for a weak duality and a strong duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem. The results on robust duality theorems are established in terms of underestimators. Additionally, to illustrate or support this study, some examples are presented.


1.
Introduction. It has been acknowledged that the data of objective function or constraints of the numerous practical programs, including real-world optimization problems, are usually not known exactly beforehand and infrequently uncertain owing to errors (from estimation, prediction or measurement) and asymmetric information [3,4]. Solving the optimization problems with ignorance of the uncertainty may result in solutions which are suboptimal for even infeasible, so it is essential to take care of uncertain optimization problems, which have data uncertainty within the objective function and/or the constraint; see, e.g., [4,5,6,7,8,9,10,11]. In line with this fact, it is imperative to review the optimization problems with data uncertainty [3,8,12,13,14]. A robust optimization, which is its robust counterpart of an uncertain optimization problem [3], has emerged as a useful deterministic approach in [5,10] to study optimization problems with data uncertainty. The concept of the robust counterpart is minimizing the value of the objective function in the worst case of all scenarios and getting a solution that works well even in the worst-case scenario but is also immunized against the data uncertainty [11]. Furthermore, constraints are enforced for every possible value of the parameters within their prescribed uncertainty set.
On the other hand, seeking a global solution is of fundamental importance in mathematical programming, where problems have many, often infinitely many, possible solutions. Nevertheless, it is known that locating a global solution of a multiextremal problem, whose several local solutions are not global, is intrinsically tough [15,16,17,18]. Accordingly, improving the criteria for identifying global solutions of optimization problems is of noticeable interest. Over the years, much attention has been targeted on developing criteria for identifying global solutions of nonconvex optimization problems [19,20,21]. The KKT condition, which is, under a certain qualification, a necessary condition for a feasible point to be a locally optimal solution of a mathematical programming problem, is often used to creating criteria for identifying global solutions. For instance, a feasible solution that fulfills the KKT conditions, referred to likewise as the KKT point, is a global minimizer for a convex programming problem. Different criteria for identifying global minimizers of nonconvex programming problems regarding the KKT conditions have been given in the literature. Nonetheless, a significant part of work in this area the KKT sufficiency frequently necessitates that the Lagrangian associated with the problem fulfills certain generalized convexity conditions [22,23,24,25,26,27].
In [1], the authors employed underestimators of the Lagrangian functions for developing criteria for a KKT point to be a global minimizer of a mathematical programming problem as well as presented also sufficient conditions for weak and strong duality results in terms of underestimators. Furthermore, in [2], the authors presented geometric criteria, which is established in terms of underestimators of the Lagrangian of a mathematical programming problem, for a KKT point to be a global minimizer of such problem with or without bounds on the variables. The idea of dealing with a problem by requiring its Lagrangian to admit underestimator with certain generalized convexity conditions, which was given [1,2], is very effective and interesting. It leads us to interesting : "How does one obtain sufficient optimality conditions for a robust KKT point to be a global robust optimal solution of a smooth nonconvex optimization problem with data uncertainty in terms of underestimators?" and also, How does one derive robust duality results between the uncertain optimization problem and its dual uncertain optimization problem?". These questions motivate us to handle with a smooth nonconvex uncertain optimization problem that features with data uncertianty in both objective and constrains for obtaining global optimality conditions and duality theorems for robust optimal solutions.
In this paper, we employ underestimators of the Lagrangian function for developing criteria for a robust KKT point to be a global robust optimal solution of a smooth nonconvex optimization problem with data uncertainty in both objective function and constraints. We establish sufficient optimality conditions by requiring an underestimator, rather the Lagrangian, to satisfy a geometric condition. Here, we present that a robust KKT point is a global robust optimal solution if the Lagrangian associated with the uncertain problem admits an underestimator that fulfills certain conditions (the convexity, or the pseudo-convexity, or the property that every stationary point is a global minimizer). In special, by using the fact that the biconjugate function of the Lagrangian is a convex underestimator at a point whenever it coincides with the Lagrangian at that point, we derive the sufficient optimality conditions as well. We also obtain results on sufficient conditions for a robust weak duality and a robust strong duality between the smooth uncertain optimization problem and its conventional Wolf type dual optimization problem, by proving the sufficient conditions for a weak duality and a strong duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of it conventional Wolf type dual model, in terms of underestimators of the Lagrangian function.
The paper is organized as follows. In section 2, we recall some notions and give some preliminary results. In section 3, several sufficient optimality conditions for a robust KKT conditions to be a global robust optimal solution in terms of underestimators are presented, and also, some examples for illustrating the applications of our results to nonconvex optimization problem with data uncertainty are given. In section 4, we present some results on sufficient conditions for a robust weak duality and a robust strong duality between the smooth uncertain optimization problem and its uncertain dual problem. Finally, in section 5, we conclude our study.

2.
Preliminaries. Let us first recall some notation and preliminary results which is able to used throughout this paper. First of all, let R n , n ∈ N, be the n-dimensional Euclidean space and for x, y ∈ R n , the notation x T y stands for the inner product of x and y.
Let f : R n → R be a real-valued function, it is said to be continuously differentiable if the derivative f (x), where x ∈ R n , exists and is itself a continuous function. Besides, a point x ∈ R n is called a stationary point of the function f : R n → R if f is differentiable at x and ∇f (x) = 0, where ∇f denotes the usual gradient of f. Simultaneously, the function f is said to be a convex function if for any x, y ∈ R n and λ ∈ [0, 1], For a convex function, its local minimizer is also a global minimizer, and if the convex function is differentiable, then its stationary point is its global minimizer as well. Furthermore, let C be a given subset of R n and the function f be (Gâteaux) differentiable at a point x ∈ C, then f is said to be pseudoconvex at x if The term pseudoconvex is employed to explain the very fact that such functions share many properties of convex functions, particularly with regards to derivative properties and finding local extrema. Note, however, that pseudoconvexity is strictly weaker than convexity as every convex function is pseudoconvex though one easily checks that f (x) = x + x 3 is pseudoconvex and non-convex. The Legendre-Fenchel conjugate function (for short, conjugate function) of f : for all x ∈ R n . The function f * is lower semicontinuous convex irrespective of the nature of f but for f * to be proper, we need f to be a proper convex function. By reiterating the operation f → f * * on f, we get the biconjugate of f, defined for all Clearly, if f is identically equal to +∞, then f * * is identically equal to −∞. Note that f * * : R n → R ∪ {−∞} is a convex function and it coincides with f when f is proper, lower semicontinuous and convex. Next, we recall the conceptbs of underestimators, introduced in [1, 2] for developing conditions for a KKT point to be a global minimizer of a standard mathematical programming problem. First of all, please consider the following mathematical programming problem: where Ω is a closed and convex subset of R n and f, g i : . . , m} and let the feasible set K := Ω ∩ A.
The following conceptbs of underestimators for developing conditions for a KKT point to be a global minimizer of (P) can be found in [1,2].
is an underestimator of f atx over K and it attains its minimum over K atx; (iii) a smooth underestimator of the function f : R n → R atx over the feasible set K if it is an underestimator of f atx over K that is differentiable atx and ∇f (x) = ∇ρ(x).
It is easy to check from the definition of biconjugate function of f that, for each , then we get from Definition 2.1 that f * * is a convex underestimator of f atx over R n .
A feasible pointx ∈ K is said to be a local minimizer of (P) if there is a neighborhood U ofx such that for each x is called a global minimizer of (P). If a feasible pointx is a local minimizer of (P) and if a certain constraint qualification holds then the following KKT conditions hold atx with multiplier λ ∈ R m + : is the Lagrangian associated with (P) and R m + is the set of all nonnegative vectors of R m .
To conclude this section, we recall some important conceptions which will be used in the sequel. Consider the following parameterized program, which is an analog of the deterministic program (P) if the objective, as well as the constraints, are uncertain: Here u and v i are uncertain parameters belonging to compact convex uncertainty sets U ⊆ R p , and V i ⊆ R q , i = 1, . . . , m, respectively. f : R n × R p → R and g i : R n × R q → R are continuously differentiable function on an open subsets of R n × R p , and R n × R q , respectively. By enforcing the constraints for all possible uncertainty within V i , i = 1, . . . , m, the problem (UP) becomes an uncertain semiinfinite program, which is the robust (worst-case) counterpart of (UP): Minimize max In section 3 and section 4, we investigate the smooth uncertain programming problem (UP) by using examine (RP). The following definition is of the set, is termed as the robust feasible set of (UP).
The set of all robust feasible solutions of (UP), equivalently the set of all feasible solution of (RP), is called the robust feasible set of (UP) and is defined by To avoid triviality in (UP), we always assume that K = ∅.
The below notion commonly referred to as robust optimal solution or robust minimax solution, can be found in [3]. This concept has been studied extensively by many authors, see, e.g., [8,12]. Definition 2.3. A robust feasible solutionx ∈ K is said to be a local robust optimal solution for (UP) if it is a global optimal solution of (RP), i.e., there exists a neighborhood U ofx such that for each x ∈ K ∩ U, Specially, if U = R n , thenx ∈ K is said to be a global robust optimal solution for (UP), equivalently, a global optimal solution of (RP).
The Lagrangian associated with (UP), denoted by L(x, u, v, λ), is given as follows: Ifx is a local robust optimal solution of (UP), i.e., a local optimal solution of (RP), and if a certain constraint qualification holds at that point, then the following robust KKT conditions hold: ∃ū ∈ U, ∃v ∈ V, ∃λ ∈ R + m such that . . , m. Whenever the robust KKT conditions hold at a robust feasible solutionx ∈ K, the pointx is said to be a robust KKT point.
3. The robust KKT sufficiency for global robust optimal solutions via underestimators. In this section, we establish that a robust KKT point is a global robust optimal solution of (UP) if the Lagrangian associated with (UP) admits an underestimator, which is convex or, more generally, has the property that every stationary point is its global minimizer. In addition, by using the fact that the biconjugate function of the Lagrangian is a convex underestimator at a point whenever it coincides with the Lagrangian at that point, we establish sufficient optimality conditions for the point to be a global robust optimal solution of (UP).
The following lemmas are used for obtaining our later results.
Lemma 3.1. Letx ∈ K and the robust KKT conditions hold atx with the uncertain parametersū,v and the multiplierλ. Suppose that the Lagrangian L(·,ū,v,λ) associated with (UP) admits a minimizing underestimator atx over the robust feasible set K. Then,x is a global robust optimal solution of (UP).
Proof. Let ρ be a minimizing underestimator of L(·,ū,v,λ) atx over K and let x ∈ K be arbitrary. Then, for the uncertain parametersū,v and the multiplierλ, Since x was arbitrary, this inequality yields max u∈U f (x, u) ≥ max u∈U f (x, u) for all x ∈ K. Thus,x is a global optimal solution of (RP), and so it is a global robust optimal solution of (UP) as desired.
The following theorem shows that every smooth underestimator atx over Ω, which is pseudo-convex, is a minimizing underestimator at that point over Ω.
Theorem 3.2. Letx ∈ K and the robust KKT conditions hold atx with the uncertain parametersū,v and the multiplierλ. Suppose that ρ : R n → R is a smooth underestimator of Lagrangian L(·,ū,v,λ) associated with (UP) atx over Ω and ρ is pseudo-convex atx over Ω. Then, the function ρ is a minimizing underestimator of L(·,ū,v,λ) atx over Ω as well asx is a global robust optimal solution of (UP).
Proof. By the definition of the smooth underestimator of a function at a point, we have ∇ρ(x) = ∇L(x,ū,v,λ). Hence, from the robust KKT conditions, we have It then follows from the pseudo-convexity of ρ that ρ(x) ≥ ρ(x), for all x ∈ Ω. This means ρ is a minimizing underestimator of L(·,ū,v,λ) associated with (UP) atx over K ⊆ Ω. Then, by Lemma 3.1, the pointx is a global robust optimal solution of (UP). Lemma 3.3. If ρ is a convex underestimator of L(·,ū,v,λ) associated with (UP) atx over Ω and ρ is differetiable atx, then ρ is a pseudo-convex underestimator of L(·,ū,v,λ) atx over Ω.
Proof. Since ρ is convex over Ω and is differentiable atx, we have from the gradient inequality that . This means ρ is pseudo-convex atx over Ω, and so the conclusion holds.
The following result follows from Theorem 3.2 and Lemma 3.3. Corollary 1. Letx ∈ K and the robust KKT conditions hold atx with the uncertain parametersū,v and the multiplierλ. Suppose that ρ : R n → R is a smooth convex underestimator of Lagrangian L(·,ū,v,λ) associated with (UP) atx over Ω. Then, the function ρ is a minimizing underestimator of L(·,ū,v,λ) atx over Ω as well as x is a global robust optimal solution of (UP).
Proof. Since ρ is a smooth underestimator of L(·,ū,v,λ) atx over Ω, it is differentiable atx. Then, from Lemma 3.3, ρ is pseudo-convex atx over Ω. Hence, it follows from Theorem 3.2 thatx is a global robust optimal solution of (UP).
The following results (Theorem 3.4, Corollary 2, Corollary 3 and Corollary 4) are applicable for the problem (UP) with Ω = R n . The sufficient conditions for a robust KKT point to be a global robust optimal solution of (UP) are presented in the following theorem.
Theorem 3.4. For the uncertain optimization problem (UP), let Ω = R n . Let x ∈ K and the robust KKT conditions hold atx with the uncertain parameters u,v and the multiplierλ. Suppose that the Lagrangian L(·,ū,v,λ) associated with (UP) admits an underestimator atx and this underestimator is differentiable atx. If every stationary point of the underestimator is its global minimizer, thenx is a global robust optimal solution of (UP).
Proof. Let ρ be an underestimator of L(·,ū,v,λ) atx, and let d ∈ R n be arbitrary. Then, for each x ∈ R n , ρ(x) ≤ L(x,ū,v,λ). So, for any µ > 0, we have Taking µ → 0, we obtain ∇ρ(x) T d ≤ ∇L(x, ·,ū,v,λ) T d = 0. Hence, ∇ρ(x) ≤ 0 for each d ∈ R n . Therefore, ∇ρ(x) = 0, and so the pointx is a stationary point of ρ. By the assumption, it then follows that, ρ(x) ≥ ρ(x) for all x ∈ R n and hence for all x ∈ K ⊆ R n , ρ(x) ≥ ρ(x) as well. Clearly, ρ is an underestimator of L(·,ū,v,λ) atx over K and it attains its minimum over the set K atx. That is to say, ρ is a minimizing underestimator of L(·,ū,v,λ) atx over R n and so, by Lemma 3.1,x is a global robust optimal solution of (UP).
Let us give some examples of multi-extremal nonconvex optimization problems that have some local robust optimal solutions such that the robust KKT conditions hold at each of the solutions. In each example, the Lagrangian admits a nonconvex underestimator at only one local robust optimal solution with the property that every local minimizer of the underestimator is its global minimizer. Example 1. Consider the nonconvex problem: where u ∈ U := [1,2], v ∈ V := [1,2]. The robust counterpart of (UP 1 ) is the following robust optimization problem: It is not hard to see that the set of all robust feasible solusions of (UP 4]. Besides, the robust optimization (RP 1 ) can be written as follows: otherwise.
Then, 0 and 3 2 are local robust optimal solutions of (UP 1 ) and the robust KKT conditions are satisfied at both points withū = 1,λ = 0 and anyv ∈ V. Furthermore, the Lagrangian L(·,ū,v,λ) associated with the problem (UP 1 ) is given by 16 ; otherwise is a (nonconvex) underestimator of the Lagrangian L(·,ū,v,λ) atx = 3 2 . In addition, we also obtain otherwise. It can be seen that, all the stationary points of ρ are its global minimizers and the pointx = 3 2 is a global robust optimal solution of (UP 1 ). The following example is motivated by Example 2.1, in [1], whose objective and constraint functions are absent of data uncertainty.

Corollary 2.
For the uncertain optimization problem (UP), let Ω = R n . Letx ∈ K and the robust KKT conditions hold atx with the uncertain parametersū,v and the multiplierλ. If the Lagrangian L(·,ū,v,λ) associated with (UP) admits a convex underestimator atx, thenx is a global robust optimal solution of (UP).
Proof. Let ρ be the convex underestimator of L(·,ū,v,λ) atx over R n , and let d ∈ R n be arbitrary. Since for each x ∈ R n , ρ(x) ≤ L(x,ū,v,λ), we obtain that for each µ > 0, By taking µ → 0, we arrive ρ (x, d) ≤ ∇L(x,ū,v,λ) T d = 0. Since ρ is convex and ρ (x, d) ≤ 0 for all d ∈ R n , ρ is differentiable atx and ∇ρ(x) = 0. Thus,x is a stationary point of the convex function ρ and so it is a global minimizer of ρ over R n . Hence, ρ is a minimizing underestimator of L(·,ū,v,λ) atx over K ⊆ R n and then it follows from Lemma 3.1 thatx is a global robust optimal solution of (UP).
The following examples illustrate Corollary 2.
Example 3. Consider the nonconvex uncertain optimization problem (UP 1 ) defined as in Example 1. The function ρ : R → R defined by It can be seen that, the point x = 3 2 is a global robust optimal solution of (UP 1 ).

Corollary 4.
For the uncertain optimization problem (UP), let Ω = R n . Letx ∈ K and the robust KKT conditions hold atx with the uncertain parametersū,v and the multiplierλ. If the Lagrangian L(·,ū,v,λ) associated with (UP) is convex, thenx is a global robust optimal solution of (UP).
Proof. Clearly, by assumption, L(·,ū,v,λ) admits its convex underestimator atx. Hence, it follows from Corollary 2 thatx is a global robust optimal solution of the problem (UP).

Remark 2.
In the special case that there is no uncertainty in the objective function and constraint function, i.e., U and V are singletons, it can be seen easily that 4. Duality. In this section, we study particularly the case that Ω of (UP) is R n . By virtue of robust optimization, we formulate a Wolfe type robust dual problem for the uncertain optimization problem and then discuss the robust weak duality and robust strong duality properties for a class of, not necessarily convex, possibly multi-extremal, optimization problems. To begin, let us call the following uncertain optimization problem (UP) with Ω = R n as (UP 0 ): The robust counterpart of (UP 0 ) is the following problem: Let y ∈ R n and λ ∈ R m + . For each fixed u ∈ U and v ∈ V, the conventional Wolfe type dual program of (UP 0 ) is given by The optimistic counterpart of the uncertain dual (UD 0 ), called optimistic dual optimization problem, is a deterministic optimization problem which is given by where the maximization is also over all the parameters u ∈ U and v ∈ V. The feasible solution set of (OD 0 ) is defined by K D := (y, u, v, λ) ∈ R n × U × V × R m ) : Remark 3. In the special case that there is no uncertainty in the objective function and constraint function, i.e., U and V are singletons, (RP 0 ) becomes min{f (x) : g i (x) ≤ 0, i = 1, . . . , m}, and (OD 0 ) collapses to the following Wolfe type dual problem: max{L(y, λ) = f (y) + m i=1 λ i g i (y) : ∇L(y, λ) = 0, λ i ≥ 0, i = 1, . . . , m}. It is also worth to note that by Wolfe type robust strong duality we understand the situation when the optimal value of (RP 0 ) equals the optimal value of (OD 0 ) and both the minimum of (RP 0 ), denoted by min (RP 0 ), and the maximum of (OD 0 ), denoted by max (OD 0 ), are attained.
GLOBAL OPTIMALITY CONDITIONS AND DUALITY THEOREMS...

105
Taking µ → 0, we obtain ( L) ((y, u, v, λ), d) ≤ ∇L(y, u, v, λ) T d = 0. It then follows from the convexity of L and ( L) ((y, u, v, λ), d) ≤ 0, for each d ∈ R n that L is differentiable at (y, u, v, λ) and ∇ L(y, u, v, λ) = 0. That is (y, u, v, λ) is a stationary point of L. Since every stationary point of a convex function is its global minimizer, we obtain L(y, u, v, λ) ≤ L(z, u, v, λ) for all z ∈ R n . Then, following the scheme for proving the desired inequality in Theorem 4.1, we arrive the conclusion of this corollary.
Proof. Letx be a robust of optimal solution of (UP 0 ), then it an optimal solution of (RP 0 ). Since the KKT conditions hold atx, there existū ∈ U,v i ∈ V i and λ i ≥ 0, i = 1, . . . , m such that f (x,ū) = max u∈U f (x, u), Hence, the conclusion of this theorem holds.
Proof. The scheme for proving this corollary is the same as in the proof of Theorem 4.2, but Corollary 5 is used instead of Theorem 4.1.
Proof. The conclusion of this corollary follows easily from Corollary 7 and Theorem 4.2.

5.
Conclusion. In this paper, a robust optimization problem, which has a maximum function of continuously differentiable functions as its objective function, is investigated. We present new conditions for a robust KKT point to be a global robust optimal solution of such uncertain optimization problems which may have many local robust optimal solutions that are not global. The obtained conditions make use of underestimators, which were first introduced by Jeyakukar and Srisatkunarajah [1,2] of the Lagrangian at that robust KKT point. We also investigate Wolfe type robust duality between the smooth uncertain optimization problem and its uncertain dual problem by proving the sufficient conditions for the weak and strong duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem. The obtained conditions for duality results are established in terms of underestimators. Also, to illustrate or support this study, some examples are presented.