Existence of Generalized Augmented Lagrange Multipliers for Constrained Optimization Problems

: The augmented Lagrange multiplier as an important concept in duality theory for optimization problems is extended in this paper to generalized augmented Lagrange multipliers by allowing a nonlinear support for the augmented perturbation function. The existence of generalized augmented Lagrange multipliers is established by perturbation analysis. Meanwhile, the relations among generalized augmented Lagrange multipliers, saddle points, and zero duality gap property are developed.

The classical Lagrangian function for the problem (P) is defined as A non-zero duality gap maybe arise for nonconvex optimization problems when using the above Lagrangian functions. Hence some modifications are necessary to overcome this difficulty, such as the augmented Lagrangian by introducing an augmented term, or the nonlinear Lagrangian by replacing the multiplier item and augmented term together by a nonlinear function. For example, the Hestenes-Powell-Rockafellar augmented Lagrangian [1][2][3], the cubic augmented (P (y,z) ) min x∈Ω f (x) s.t. g i (x) + y i ≤ 0, i = 1, · · · , m, h j (x) + z j = 0, j = 1, · · · , l.
Denote by val(P) and v(y, z)(:= val(P (y,z) )) the optimal values of (P) and (P (y,z) ), respectively. Clearly, v(0, 0) = val(P). Denote by X * the optimal solution set of problem (P), and assume throughout the paper that the optimal value val(P) is finite.
Here σ is called an augmenting function (see Section 2 below for details). Its properties are weakened from convex to level-bounded, or valley-at-zero. For example, in Rockafellar and Wets [34], a nonnegative convex augmenting function and the corresponding augmented Lagrangian dual problem of primal problem were introduced. A sufficient condition for the zero duality gap and a necessary and sufficient condition for the existence of an exact penalty representation were obtained. It was extended in [35] by replacing the convexity condition of the augmenting function with a level-boundedness condition. Using the theory of abstract convexity, a family of augmenting functions with almost peak at zero property and a class of corresponding augmented Lagrangian dual problems were introduced in [36]. Valley-at-zero property (similar to almost peak-at-zero property) was used in [37]. A vector (λ, µ) is said to be an augmented Lagrange multiplier for problem (P) (cf. [22,25]), if v r (y, z) ≥ v r (0, 0) + λ, y + µ, z , ∀(y, z) ∈ R m × R l .
That means that (λ, µ) is a subgradient of v r (·, ·) at (y, z) = (0, 0). The set of all subgradients (λ, µ) is called the subdifferential of v r (y, z) at (y, z) = (0, 0) and denoted by ∂v r (0, 0). Augmented Lagrange multipliers are an important concept in duality theory. Their existence is important for the global convergence analysis of primal-dual type algorithms based on the use of augmented Lagrangians [7,19,29,32,33]. In addition, augmented Lagrange multipliers are closely related to saddle points, the zero duality gap property, and exact penalty representation. Some results on the existence of augmented Lagrange multipliers are discussed for semi-infinite programming [25], cone programming [22,23], and eigenvalue composite optimization problems [38]. Moreover, CQ-free duality was proposed in the classical monograph [39] by Bonnans and Shapiro. The stronger results on CQ-free strong duality for semidefinite and general convex programming can be found in [40,41], and in more recent publications for semi-infinite, semidefinite, and copositive programming by Kostyukova and others [42,43]. Recently, Dolgopolik [44] studied the existence of augmented Lagrange multipliers for geometric constraint optimization by using the localization principle.
Recall that for convex programming, Lagrangian multiplier is a subgradient of perturbation function v at u = 0 in the sense of convex analysis; i.e., For nonconvex programming, the Lagrangian multiplier can be used to estimate the subdifferential of the perturbation function at the origin. Precisely, for a minimization problem where f : R n → R, g : R n → R m , X is a closed set in R n , and θ : R m →R := (∞, +∞] is proper, lsc, and convex. This model includes the constrained optimization problems (by letting θ be a indicator function) and composite optimization problems. Denote by S * the solution set. Forx ∈ S * , let It should be pointed out that the subdifferential that appeared in (3) is the limiting/Mordukhovich subdifferential, not a subdifferential in the sense of convex analysis. Here M ∞ (x) = {0} can be regarded as constraint qualification. In particular, if domθ := R l − × {0} l and X := R n , then this condition is Mangasarian-Fromovitz constraint qualification; if domθ is a convex cone with nonempty interior and X := R n , then this condition is Robinson's constraint qualification. The result (3) indicates that the Lagrangian multiplier provides an upper bound on the subdifferential of perturbation function and gives an estimate on the Lipschitz constant of perturbed function. It is very important for the convergence analysis of numerical algorithms.
Compared with the classical Lagrangian function, the augmented Lagrangian function has been successfully applied to study nonconvex programming. Hence an interesting question is how to use the augmented Lagrangian multiplier to study the subdifferential of v τ , and further give an estimate on Lipschitz constant on v τ . On subdifferentiability in nonconvex setting, Clarke's pioneering work on generalized gradient opened the door to the study of general nonsmooth functions. Many concepts were introduced in the past few decades. Frequently used concepts include limiting/Mordukhovich subdifferential, Ioffe's approximate and G-subdifferential, Michel and Penot's subdifferential, Treiman's linear subdifferential, Sussmann's semidifferential, etc. Compared with the abstract subdifferential (pioneered by Warga), which is defined by a set of axioms, many subdifferentials have reasonable geometric explanations. For example, a convex subdifferential means a linear support, Frechét subdifferential means a smooth support, and a proximal subdifferential means a local quadratic support. The detailed discussion on other subdifferentials and their properties (particularly on calculus rules and the robust property) can be found in [34].
Clearly, the definition of an augmented Lagrangian multiplier given in (2) indicates that the augmented perturbation function is supported by a linear function at the origin. It corresponds to the subdifferential in the convex analysis. However, for a nonconvex setting, it is natural to consider whether a nonlinear support is available. Once it is done, we can establish and apply the duality theory in a more flexible environment. Define ω : R + → R + such that ω(η) → +∞ as η → +∞. Definition 1. A vector (λ, µ) is said to be a generalized augmented Lagrange multiplier of (P), if there exists where φ i for i = 1, 2 possesses the following properties: whenever (y, z) satisfies y + g(x) ≤ 0, z + h(x) = 0, and η > 0 is sufficiently large.
Since φ i includes the inner product as special cases, (4) is an essential extension of (2) from linear support to nonlinear support.
As mentioned above, the augmented Lagrange multiplier is a subgradient (in the sense of convex analysis) of an augmented perturbation function at the origin. That means the augmented perturbation function has a linear support. The augmented Lagrange multiplier is extended in this paper to a new concept called the generalized augmented Lagrangian multiplier, in which a nonlinear support is allowed. The main aim of this paper is to study the existence of generalized augmented Lagrange multipliers. It helps us to better understand properties of an augmented perturbation function at the origin. Based on this nonlinear support, we need to re-investigate the corresponding duality theory, particularly be discussing the relations among generalized augmented Lagrange multipliers, saddle points, and the zero duality gap property. The existence of generalized augmented Lagrange multipliers is established by perturbation analysis of the primal problem.
We organize our paper as follows. Section 2 introduces the preliminaries. In Sections 3, we present the duality theory based on generalized augmented Lagrangians. Section 4 discusses the existence of generalized augmented Lagrange multipliers by perturbation analysis.

Preliminaries
In this section we clarify the notation, recall some background materials we need from duality theory, and develop some preliminary results.
where σ : R m+l → R + := [0, +∞) satisfies the following valley-at-zero property: The definition of the growth condition defined below was introduced in [23], as an extension of the one given in [3], where the augmenting function is restricted to be a quadratic function.

Definition 2.
A function v(y, z) is said to satisfy the growth condition with σ, if for any where B R m+l denotes the closed unit ball in R m+l .
The dualizing parametrization function of the primal problem is defined as For (x, λ, µ) ∈ R n × R m + × R l , the corresponding generalized augmented Lagrangian is The generalized Lagrangian function is defined as which reduces to the classical Lagrangian of (P) when φ 1 (λ, y) = λ, y and φ 2 (µ, z) = µ, z . If in particular x ∈ Ω, the generalized augmented Lagrangian can be rewritten as where the inequality comes from (A 2 ).
If the above inequalities hold for all x ∈ B R n (x * , δ) ∩ Ω, where B R n (x * , δ) denotes the ball with center x * and radius δ > 0, then (x * , λ * , µ * ) is said to be a local saddle point of L.
The generalized augmented Lagrangian dual problem of (P) is defined as where θ(λ, µ, r) is the generalized augmented Lagrangian dual function given as Taking into account of (7) and (10), we have In addition, it also follows from (5) that It is well known that a zero duality gap between the problem (P) and its generalized augmented For r ≥ 0, consider the following r-dual problem of (P), denoted by (D r ), Similarly, if for some fixed r ≥ 0 such that then the zero duality gap property holds for the pair of problems (P) and (D r ).
Define the optimal values of problems (D) and (D r ) by val(D) and val(D r ), respectively. It is clear that val(D) = sup r∈R + val(D r ).

Duality Theory Based on Generalized Augmented Lagrangian Functions
In this section, we study the relationships among generalized augmented Lagrange multipliers, global saddle points, and the zero duality gap property between the primal problem and its generalized augmented Lagrangian dual problem. The related conclusions are given in Theorem 3 and Theorem 4.
Firstly, the weak duality theorem is given below, which shows that the dual problem provides a lower bound for (P). Proposition 1. Let x be a feasible point of (P) and (λ, µ, r) ∈ R m + × R l × R + . Then where the inequality follows by letting y = 0, z = 0 and φ(·, 0) = 0. Hence The arbitrariness of x ensures is a generalized augmented Lagrange multiplier of (P) with r * if and only if (λ * , µ * , r * ) is an optimal solution of (D) and the zero duality gap property holds for problems (P) and (D).
From the proof of Theorem 1, we can see that (λ * , µ * ) is an optimal solution of (D r * ) and the zero duality gap property holds between (P) and (D r * ). It should be emphasized that the existence of generalized augmented Lagrange multipliers does not require that the primal problem (P) must be solvable. Indeed, in general, the optimal solution of a primal problem cannot be known in advance. The relation between the zero duality gap property and global saddle points is given below. Theorem 2. Let σ : R m+l → R + and r * ≥ 0. Then (x * , λ * , µ * ) is a global saddle point of L(x, λ, µ, r * ) if and only if val(P) = val(D r * ), and x * ∈ Ω, (λ * , µ * ) ∈ R m + × R l are optimal solutions of (P) and (D r * ), respectively.
Proof. We first claim that Consider the following two cases: (5) and (6) we get If x ∈ Ω, but x / ∈ F , it follows from the property (A 3 ) that there exist nonzero (λ 0 , µ 0 ) ∈ R m + × R l and γ < 0 such that whenever (y, z) satisfies y + g(x) ≤ 0, z + h(x) = 0, and η > 0 sufficiently large. Hence where the first inequality comes from the nonnegativity of σ, and the second inequality is due to (16). This together with γ < 0 further implies that i.e., sup Therefore, either x / ∈ Ω or x ∈ Ω, x / ∈ F , so it follows from (15) and (17) that Case 2.
x is feasible i.e., x ∈ Ω and g(x) ≤ 0, h(x) = 0. In this case, it follows from (12) that for According to the nonnegativity of σ, we also have which together with (19) means that Putting (18) and (20) together yields the desired formula (14). Hence On the other hand, note that the dual problem can be rewritten as The desired result follows by applying the minimax relations theorem (Theorem 11.50 [34]).
Indeed, Theorem 2 shows that val(P) = val(D), and x * , (λ * , µ * , r * ) are optimal solutions of (P) and (D) respectively, provided that val(D) = val(D r ), i.e., val(D) = sup r∈R + val(D r ) by Proposition 1, and the maximum can be attained at some r. The converse statement obviously holds true. As just mentioned above, compared with the existence of augmented Lagrange multipliers, global saddle points require that the primal problem is solvable. Theorem 3. Suppose that σ : R m+l → R + has a valley at zero, v satisfies the growth condition with σ, and lim inf (y,z)→(0,0) v(y, z) < +∞. (21) The following statements hold: (ii) v is lower semi-continuous at the origin if and only if the zero duality gap property holds for problems (P) and (D).
(ii). If v is lower semi-continuous at origin, then lim inf Therefore, the zero duality gap property holds for (P) and (D). Conversely, according to (22), it is easy to see that the lower semi-continuity of v at the origin can be obtained if the zero duality gap property holds for problems (P) and (D). Corollary 1. Suppose that σ : R m+l → R + has a valley at zero, v satisfies the growth condition with σ, and lim inf (y,z)→(0,0) v(y, z) < +∞.
Proof. The results follow immediately from Theorem 3.
Theorem 3 shows that the zero duality gap property is closely related with the lower semi-continuity of the perturbation function. In the definition of generalized augmented Lagrange multipliers, the inequality involved in (4) is required to be satisfied for all (y, z) ∈ R m+l , but Theorem 4 shows that this restriction can be weakened by just checking all (y, z) in some neighborhood of the origin once some additional assumptions are imposed on augmented functions. In the following, we further require the φ satisfying the following property: (A 4 ) For any (λ, µ) ∈ R m + × R l , there exist ρ > 0, τ > 0 such that Theorem 4. Suppose that σ has a valley at zero and v satisfies the growth condition with σ. Then (P) has a generalized augmented Lagrange multiplier (λ * , µ * ) ∈ R m + × R l if and only if there exists r * ∈ R + such that Proof. (Necessity). The necessity is clear by the definition of generalized augmented Lagrange multiplier.
(Sufficiency). Since v satisfies the growth condition with σ, then for any Since σ has a valley at zero, there exists d > 0 such that Combining the property (A 4 ) with (31) means that for any (y, It follows from (29) and (32) that Hence (λ * , µ * ) is a generalized augmented Lagrange multiplier of (P).
Here we list two classes of nonlinear functions satisfying the above assumptions (A 1 )-(A 4 ).
(2) Let θ : R + → R satisfy θ(t) > 0 if t = 0 and θ(t) ≥ t q as t > 0 are sufficiently large, where q is positive integer. Define where A is a symmetric and invertible matrix.

Existence of Generalized Augmented Lagrange Multipliers
In this section, we develop some sufficient conditions for the existence of generalized augmented Lagrange multipliers. Given ε ≥ 0, define Lemma 1. Suppose that σ : R m+l → R + has a valley at zero and Then for any ε > 0, we have lim L(x, λ * , µ * , r) = +∞, (34) and whenever r > 0 is sufficiently large.
Proof. The proofs of (34) and (35) are given in parts (a) and (b), respectively.
(a) For any fixed x ∈ Ω/W 1 (ε), it follows from the definition of which implies that for any (ξ 1 , According to the valley-at-zero property of σ, for any x ∈ Ω/W 1 (ε), there exists ζ > 0 such that It follows from (8) that where the second inequality comes from (36). This implies that Passing to limit in the above inequality, we get where the equality comes from the fact that inf x∈Ω L 0 (x, λ * , µ * ) − sup ξ 1 ≤0 φ 1 (λ * , ξ 1 ) is finite by (33). Hence, (34) is true.
Suppose, on the contrary, that there exist ε 0 > 0, r k → ∞ and x k ∈ Ω such that From (7) and (37), we conclude that there exist By the property (A 4 ), for above (λ * , µ * ), there exist ρ > 0, τ > 0 such that Further using valley-at-zero property of σ, for above τ > 0 there exists d 1 > 0 such that Now, let us prove that (y k , z k ) → 0. Let us consider the following cases: Case 1. There exists an infinite subset N 1 ⊆ N such that (y k , z k ) ≥ τ for all k ∈ N 1 . Note that where the second inequality comes from the assumption (39), and the third step is due to (40). The right side in (41) can be arbitrary large as N 1 k → ∞, which contradicts the finiteness of v(0, 0). Case 2. (y k , z k ) ≤ τ as k sufficiently large. Since φ 1 and φ 2 are continuous by the property A 1 , for above (λ * , µ * ) and τ > 0, we can find d 2 ∈ R such that Then Due to the boundedness of (y k , z k ), we have which in turn implies that (y k , z k ) → 0 by the valley-at-zero property of σ. Applying (y k , z k ) → 0 into (38) yields v(0, 0) + ε 0 2 ≥ f (x k ). It contradicts x k / ∈ W 2 (ε 0 ) by the definition of W 2 (ε). Therefore, (35) holds.
Proof. According to the relationship among the generalized augmented Lagrange multiplier, the zero duality gap property, and global saddle points established in Theorems 1 and 2, we only need to justify that (x * , λ * , µ * ) is a global saddle point of L(x, λ, µ, r).
According to the definition of local saddle points, there exists δ > 0 such that It follows by invoking (14) and the first inequality in (43) that By the monotonicity of L(x * , λ * , µ * , r) in r, we also have where the second inequality comes from (12). Combining (44) and (45) implies which together with (12) again yields Now, we establish the first inequality in (9). To complete the proof, it remains to show that whenever r is sufficiently large. Suppose on the contrary that we can find r k → +∞ and Hence, applying (46) into (49) and together with the fact x * ∈ X * yields which means that x k belongs to the set {x ∈ Ω| L(x, λ * , µ * , r k ) ≤ v(0, 0)}. Taking into account of (35) in Lemma 1, we obtain that for any ε ∈ (0, ε 0 ), x k ∈ W 1 (ε) W 2 (ε), which further implies that x k ∈ Λ by (42). We can assume without loss of generality that x k converges tox. According to the continuity of f (x), g(x) and h(x), together with the closedness of W 1 (ε) and W 2 (ε), we obtain that x ∈ W 1 (ε) W 2 (ε). Therefore,x ∈ W 1 (0) W 2 (0) by the arbitrariness of ε > 0, which further implies thatx ∈ X * . By assumption, (x, λ * , µ * ) is also a local saddle point of L(x, λ, µ, r) for somer > 0; i.e., there existsδ > 0 such that Similar to the above argument, it follows from (44) that Since x k ∈ B R n (x,δ) and r k ≥r for k large enough, from (51) and (52), it follows which contradicts (50). This justifies (48). By the fact (43), (47) and (48), we conclude that (x * , λ * , µ * ) is a global saddle point of L(x, λ, µ, r) for r large enough. Therefore, (λ * , µ * ) is a generalized augmented Lagrange multiplier of (P). Theorem 6. Suppose that σ : R m+l → R + has a valley at zero and inf x∈Ω L 0 (x, λ * , µ * ) − sup Let x * be the unique global optimal solution of (P). If (x * , λ * , µ * ) is a local saddle point of L(x, λ, µ, r) for some r ≥ 0, and there exists ε 0 > 0 such that where Λ is a bounded subset in R n , then (λ * , µ * ) is a generalized augmented Lagrange multiplier of (P).
To complete the proof, we next need to show that there exists r 2 > 0 such that Suppose on the contrary that there exist r k → ∞ and {x k } ⊂ W 1 (ε 0 ) such that f (x * ) > L(x k , λ * , µ * , r k ).
The existence of generalized augmented Lagrange multipliers is established in two different scenarios: one is applicable to the case of unique solution while another is applicable to the case of multiple optimal solutions.

Conclusions
In this paper, we studied the generalized augmented Lagrangian multiplier, which is an extension of the augmented Lagrangian multiplier from linear support to nonlinear support for an augmented perturbation function. Some sufficient conditions for the existence of generalized augmented Lagrangian multipliers were developed. In particular, the relationships among global saddle points, generalized augmented Lagrangian multipliers, and the zero duality gap property between the primal problem and its generalized augmented Lagrangian dual problem were established. Several interesting topics are left for further investigation. For example, one is developing some necessary and sufficient conditions for the existence of generalized augmented Lagrangian multipliers by using the localization principle; another is studying the generalized differentiation of support functions from the subdifferential view.