On Robust Saddle-Point Criterion in Optimization Problems with Curvilinear Integral Functionals

: In this paper, we introduce a new class of multi-dimensional robust optimization problems (named ( P ) ) with mixed constraints implying second-order partial differential equations (PDEs) and inequations (PDIs). Moreover, we deﬁne an auxiliary (modiﬁed) class of robust control problems (named ( P ) ( ¯ b ,¯ c ) ), which is much easier to study, and provide some characterization results of ( P ) and ( P ) ( ¯ b ,¯ c ) by using the notions of normal weak robust optimal solution and robust saddle-point associated with a Lagrange functional corresponding to ( P ) ( ¯ b ,¯ c ) . For this aim, we consider path-independent curvilinear integral cost functionals and the notion of convexity associated with a curvilinear integral functional generated by a controlled closed (complete integrable) 1-form.


Introduction
As we all know, partial differential equations (PDEs) and partial differential inequations (PDIs) are essential in modeling and investigating many processes in engineering and science. In this respect, many researchers have taken a special interest in their study. We specify, for example, the research works of Mititelu [1], Treanţȃ [2][3][4], Mititelu and Treanţȃ [5], Olteanu and Treanţȃ [6], Preeti et al. [7], and Jayswal et al. [8] on the study of some optimization problems with ODE, PDE, or isoperimetric constraints. In order to reduce the complexity of the considered optimization problems, some auxiliary optimization problems were formulated to investigate the initial problems more easily (Treanţȃ [9][10][11][12]). Nevertheless, since the real-life processes and phenomena often imply uncertainty in initial data, many researchers have turned their attention to optimization issues governed by first-and second-order PDEs, isoperimetric restrictions, stochastic PDEs, uncertain data, or a combination thereof. In this context, we mention the following research papers: Wei et al. [13], Liu and Yuan [14], Jeyakumar et al. [15], Sun et al. [16], Preeti et al. [7], Lu et al. [17], and Treanţȃ [18]. The structure of approximate solutions associated with some autonomous variational problems on large finite intervals was studied by Zaslavski [19]. Furthermore, Geldhauser and Valdinoci [20] investigated an optimization problem with SPDE constraints, with the peculiarity that the control parameter s is the s-th power of the diffusion operator in the state equation. In [21], Babamiyi et al. focused on identifying a distributed parameter in a saddle point problem with application to the elasticity imaging inverse problem. Very recently, Debnath and Qin [22], investigated the robust optimality and duality for minimax fractional programming problems with support functions.
Motivated and inspired by previous research works, in this paper, we introduce and study new classes of robust optimization problems. More exactly, by taking curvilinear integral objective functionals with mixed (equality and inequality) constraints implying data uncertainty and second-order partial derivatives, we introduce the robust control problems under study. Further, by using the concept of convexity associated with curvilinear integral functionals and the notion of robust saddle-point associated with a Lagrange functional corresponding to the modified robust optimization problem, we formulate and prove some characterization results for the considered classes of control problems. The novelty elements included in the paper, in comparison with other research papers in this field, are provided by the presence of uncertain data both in the objective functional and in the constraint functionals and also by the presence of second-order partial derivatives. Moreover, the proofs associated with the main results are established in an innovative way. Furthermore, since the mathematical framework introduced here is appropriate for various scientific approaches and viewpoints on complex spatial behaviors, the current paper could be seen as a definitive research work for a large community of researchers in engineering and science.
The paper is structured as follows. Section 2 provides the preliminary and necessary mathematical tools, which will be used in the next sections. Section 3 includes the main results of this paper. Under convexity assumption of the cost functional, the first main result establishes a connection between a robust saddle point of the Lagrange functional associated with the associated modified problem (P) (b,c) and a weak robust optimal solution of (P). By assuming the convexity hypotheses of the constraint functionals, the converse of the first main result is presented in the second main result. In Section 4, we formulate the conclusions and further development.

Preliminaries
In this paper, we use the following working hypotheses and notations: • Consider R p , R q , R r and R n as Euclidean spaces of dimension p, q, r and n, respectively; • Consider Θ ⊂ R p as a compact domain and the point t = (t α ) ∈ Θ as a multi-parameter of evolution or multi-time; • Consider Γ ⊂ Θ as a piecewise smooth curve joining the points t 0 and t 1 in Θ; • B is the space of C 4 -class state functions b = (b τ ) : Θ → R q and b α := ∂b ∂t α , b αβ := ∂ 2 b ∂t α ∂t β denote the partial speed and partial acceleration, respectively; • C is the space of C 1 -class control functions c = (c j ) : Θ → R r ; • Consider T as the transpose for a given vector; • Consider the following convention for inequalities and equalities of any two vectors x, y ∈ R n : n and x i < y i for some i.
In the following, we consider g = (g 1 , . . . , g m ) = (g l ) : n, are C 3 -class functionals. Furthermore, let us assume that w = (w κ ), u = (u l ) and v = (v ζ ) are the uncertain parameters for some convex compact subsets W = (W κ ) ⊂ R p , U = (U l ) ⊂ R m and V = (V ζ ) ⊂ R n , respectively. Denote by J 2 Θ, R q the second-order jet bundle associated with Θ and R q . Furthermore, assume that the previous multi-time-controlled second-order Lagrangians f κ determine a controlled closed (complete integrable) Lagrange 1-form (see summation over the repeated indices, Einstein summation): which generates the following controlled path-independent curvilinear integral functional: The second-order PDE and PDI constrained variational control problem with uncertainty in the objective and constraint functionals is defined as follows: The associated robust counterpart of the aforementioned variational control problem (P) is defined as: Further, denote by the feasible solution set in (RP), and we call it the robust feasible solution set of (P).
To simplify the presentation, we use the following notation: The associated first-order partial derivatives of f κ , κ = 1, p, are defined as In the same manner, we have g b := ∂g ∂b and g c := ∂g ∂c by using matrices with m rows and h b := ∂h ∂b and h c := ∂h ∂c by using matrices with n rows.
Further, in accordance to Treanţȃ [3], we define the notion of a weak robust optimal solution of the considered class of constrained variational control problems. This notion will be used to establish the associated robust necessary conditions of optimality and the main results derived in the paper.

Definition 1.
A pair (b,c) ∈ D is said to be a weak robust optimal solution to (P) if there does not exist another Next, we shall use the Saunders's multi-index notation (Saunders [23], Treanţȃ [3,24]) to formulate the concept of convexity and the robust necessary optimality conditions for (P).
According to Treanţȃ [24], we formulate the robust necessary optimality conditions for (P).

Definition 3.
A pair (b,c) ∈ D is said to be a normal weak robust optimal solution to (P) ifμ > 0 in Theorem 1. We can considerμ = 1 without loss of generality.
Next, we use the modified objective function method to reduce the complexity of (P). In this direction, let (b,c) be an arbitrary given robust feasible solution to (P). The modified multi-dimensional variational control problem associated with the original optimization problem (P) is defined as: where the functionals g, f κ and h are given as in (P).
The associated robust counterpart of the modified multi-dimensional variational control problem (P) (b,c) is defined as:

Remark 2.
The robust feasible solution set of the problem (P) (b,c) is the same as in (P). Consequently, it is also denoted by D.

Saddle-Point Optimality Criterion
In this section, under some convexity assumptions, we establish some connections between a weak robust optimal solution of (P) and a robust saddle-point associated with a Lagrange functional (Lagrangian) corresponding to the modified multi-dimensional variational control problem (P) (b,c) . In this regard, in accordance with Treanţȃ [9,11,12] and Preeti et al. [7], we formulate the next definitions.  , c), ν, γ, w, u, v) : B × C × R m + × R n × W × U × V → R associated with the modified variational control problem (P) (b,c) is defined as ((b,c),ν,γ,w,ū,v) ∈ D × R m + × R n × W × U × V is said to be a robust saddle-point for the Lagrange functional L((b, c), ν, γ, w, u, v) associated with the modified multidimensional variational control problem (P) (b,c) if the following relations are fulfilled:

Definition 6. A point
Now, taking into account the above definitions, we establish the following two main results of this paper. Theorem 2. Let (b,c) be a robust feasible solution to (P). Assume that max w∈W f κ (π, w) = f κ (π,w), κ = 1, p, and the objective functional Γ f κ (π,w)dt κ is convex at (b,c). If the point ((b,c),ν,γ,w,ū,v) is a robust saddle-point for the Lagrange functional L((b, c), ν, γ, w, u, v) associated with the modified multi-dimensional variational control problem (P) (b,c) , then (b,c) is a weak robust optimal solution to (P).
Theorem 3. Let (b,c) be a normal weak robust optimal solution to (P). Assume that max w∈W f κ (π, w) = f κ (π,w), κ = 1, p, and the constraint functionals Proof. Since the relations (1)-(4), withμ = 1, are satisfied for all t ∈ Θ, except at discontinuities, the conditions (1) and (2) yield where we used the formula of integration by parts, the result "A total divergence is equal to a total derivative" (see Treanţȃ [4]) and the boundary conditions formulated in the considered problem. Further, taking into account the assumption of convexity for the following multiple integral functionals Γν T g(π,ū)dt κ , or, equivalently, Consequently, by (9) and (10), we conclude that ((b,c),ν,γ,w,ū,v) is a robust saddlepoint for the Lagrange functional L((b, c), ν, γ, w, u, v) associated with the modified multidimensional variational control problem (P) (b,c) , and the proof is completed.

Conclusions and Further Development
In this paper, by considering path-independent curvilinear integral cost functionals with mixed (equality and inequality) constraints implying data uncertainty and secondorder partial derivatives, we have introduced new classes of robust optimization problems. More precisely, by using the notion of convexity for curvilinear integral functionals, the concept of a normal weak robust optimal solution and the robust saddle-point of a considered Lagrange functional, we have established some characterization results of the problems under study.
As an immediate subsequent development of the results presented in this paper, the author mentions the study of well-posedness for the considered classes of robust control problems.