First- and Second-Order Optimality Conditions for Quadratically Constrained Quadratic Programming Problems

We consider a quadratic programming problem with quadratic cone constraints and an additional geometric constraint. Under suitable assumptions, we establish necessary and sufficient conditions for optimality of a KKT point and, in particular, we characterize optimality by using strong duality as a regularity condition. We consider in details the case where the feasible set is defined by two quadratic equality constraints and, finally, we analyse simultaneous diagonalizable quadratic problems, where the Hessian matrices of the involved quadratic functions are all diagonalizable by means of the same orthonormal matrix.


Introduction
We analyse a quadratic programming problem with general quadratic cone constraints and an additional geometric constraint. This problem has received attention in the literature in the last decades (see, e.g. [5,7,18,20]) since it contains as a particular case several classic optimization problems as trust region problems, the standard quadratic  Fabián Flores-Bazán fflores@ing-mat.udec.cl problem and the max cut problem; moreover, it has many applications in robust optimization under matrix norm data uncertainty and in the field of biology and economics [12]. In this paper, we are interested in establishing necessary or sufficient global optimality conditions for a point that fulfils the Karush-Kuhn-Tucker (KKT) conditions or under the assumption of strong duality on the given problem. The general formulation of the considered quadratic programming problem allows us to treat simultaneously quadratic problems with one or more quadratic equality or inequality constraints and possibly additional constraints that can be included in the geometric one, which makes the analysis of the given problem very general, particularly as regards the possibility of providing equivalent formulations and associating a dual problem with the given one. Our approach allows to recover or generalize several known results in the literature [13,14,20]. The paper is organized as follows. In Sect. 2 we recall the main definitions and preliminary results that will be used throughout the paper. In Sect. 3, we characterize global optimality for a KKT point or in the presence of the property of strong duality on the given problem and in Sect. 4, we consider in details the case where the feasible set is defined by two quadratic equality constraints. In Sect. 5 we analyse a simultaneous diagonalizable quadratic problem (S D Q P), where the Hessian matrices of the involved quadratic functions are all diagonalizable by means of the same orthonormal matrix S. The analysis previously developed allows us to provide suitable conditions that guarantee the existence of a convex reformulation of S D Q P improving some results stated in [15] in the presence of two quadratic inequality constraints.

Preliminary Results
Let us recall the basic notations and preliminary results that will be used throughout the paper. Given C ⊆ IR n , co C, int C, ri C, cl C, span C, denote the convex hull of C, the topological interior of C, the relative interior, the closure of C and the smallest vector linear subspace containing C, respectively. C is said to be a cone if tC ⊆ C, ∀ t ≥ 0. A convex cone C is called pointed if C ∩ (−C) = {0}. We define cone C := t≥0 tC. We set IR m + := {x ∈ IR m : x ≥ 0}. If C is a convex set and x ∈ C, the normal cone to C atx ∈ C is defined by N C (x) := {ξ ∈ IR n : ξ, x −x ≤ 0, ∀ x ∈ C}. The positive polar of a set C ⊆ IR n is defined by C * := {y * ∈ IR n : y * , x ≥ 0, ∀x ∈ C}. It is well known that C * = (cl C) * = (co C) * = (cone C) * , cl co(cone C) = cl cone(co C) = C * * := (C * ) * . (1) Let P ⊆ IR m be a convex cone and C ⊆ IR n a convex set. A function f : IR n → IR m is said P-convex on C if for every x 1 , x 2 ∈ C and for every λ ∈ [0, 1], For m = 1 and P = IR + , we recover the classic definition of a convex function. It is known that if f is P-convex on C, then the set f (C) + P is convex.
In the paper we will use the following preliminary results. Let C := {x ∈ IR n : g(x) = 0}, where g : IR n → IR. Then, we get [6] T (C; and so [T (C;x)] * = IR ∇g(x); whereas if g(x) . = 1 2 x Bx + b x + β is a quadratic function, with B being a real symmetric matrix of order n, b ∈ IR n and β ∈ IR, then

The General Case with Cone Quadratic Constraints
Let us consider the problem where P is a convex cone in IR m , g(x) := (g 1 (x), . . . , g m (x)) and f , g i : with A, B i being real symmetric matrices; a, b i being vectors in IR n and α, β i ∈ IR for i = 1, . . . , m. K := {x ∈ C : g(x) ∈ −P} is the feasible set of (4). We associate with (4) the Lagrangian function L(λ, x) .
λ i g i (x) and its dual problem We say that strong duality holds for (4), if there exists λ * ∈ P * such that In case (4) admits an optimal solutionx ∈ K , then the previous condition is equivalent to Under suitable assumptions on the cone T (C;x), we first establish three general results: the first and the second consider the case wherex is a KKT point and provide a sufficient optimality condition and a characterization of its optimality in the case where P = {0} m , respectively, while the third one characterizes optimality under the assumption of strong duality.
and, additionally, (K −x) ⊆ cl co T (C;x). Then the following assertion holds.
Proof By (8), ∇ x L(λ * ,x) v ≥ 0, for every v ∈ T (C;x), and by (1) we obtain The assumptions imply that We note that, since the involved functions are quadratic, then, the following equality holds: Exploiting (10) and (9), for every x ∈ K , we get By the previous inequalities, the assertion follows.
Remark 3.2 Proposition 3.1 is related to Theorem 2.1 in [4] when applied to a quadratic problem. Indeed, Theorem 2.1 in [4] requires that K is a convex set and C := IR n , which guarantees that the condition (K −x) ⊆ cl co T (C;x) is fulfilled.
. . , g m be quadratic functions as above, let P := {0} m andx ∈ K . Assume that and thatx is a KKT point for (4), i.e. there exists λ * ∈ IR m such that Then the following conditions are equivalent: (a)x is an optimal solution for the problem (4); is positive semidefinite on K −x and so on cl cone(K −x). Proof By (12), ∇ x L(λ * ,x) v ≥ 0, for every v ∈ T (C;x) and by (1) we get, The second inclusion in (11) implies that and, by the first inclusion in (11), ∇ x L(λ * ,x) v = 0, for every v ∈ (K −x). By (10) and (13), for every x ∈ K , we get By the previous equalities, the equivalence between (a) and (b) follows. (11) is not needed for proving that (b) implies (a), as shown by Proposition 3.1.

Remark 3.4 Note that the second inclusion in assumption
In the following proposition we characterize optimality under the strong duality property that can be considered as a regularity condition in view of the fulfilment of the KKT conditions. Proposition 3.5 Let f , g 1 , . . . , g m be quadratic functions as above, letx ∈ K , and assume that Then the following assertions are equivalent: (a)x is an optimal solution for the problem (4) and strong duality holds; (b) there exists λ * ∈ P * such that (8) is fulfilled and ∇ 2 x L(λ * ,x) is positive semidefinite on C −x.

Remark 3.7 Condition
For the other inclusion, take any x ∈ A. Then Remark 3.6 leads to the following result.
. . , g m be quadratic functions as above, letx ∈ K , and assume that C

is an optimal solution for the problem (4) and strong duality holds.
Proof By Proposition 3.5 and taking into account Remark 3.6, it is enough to prove that (C −x) ⊆ cl co T (C;x). The convexity of the functions g i , i = m + 1, . . . , p, yields that C is convex. Since C is convex then T (C;x) = cl cone(C −x) which implies (C −x) ⊆ cl co T (C;x) (see, e.g. [2]).
All the results so far obtained generalize optimality conditions for classical quadratic programming to a quadratic problem with cone constraints and a geometric constraint set. We now present suitable particular cases where our results allow to recover and generalize known optimality conditions. We first consider the quadratic programming problem with bivalent constraints (QP1) defined by and e j is a vector in IR n whose jth element is equal to 1 and all the other entries are equal to 0.
, be the Lagrangian function associated with (QP1). By Proposition 3.3 and Lemma 3.1 we recover Lemma 3.1 of [14] which can be stated as follows. Proposition 3.9 Let C := IR n andx ∈ K . Assume that there exist λ ∈ IR m and γ ∈ IR n such that ∇ x L(λ, γ ,x) = 0 . Thenx is an optimal solution for (QP1) if and only if ∇ 2 x L(λ, γ ,x) is positive semidefinite on Z (x) defined by (18). Proof It is enough to notice that since C = IR n , then, by Lemma 3.1, Z (x) = K −x and, moreover, (11) is fulfilled. Proposition 3.3 allows us to complete the proof.
By Proposition 3.5 we obtain the following result.
Next result is inspired by Theorem 3.1 of [14] and provides a characterization and a sufficient condition for strong duality for (QP1). Proposition 3.10 Letx ∈ K with C := IR n . Consider the following assertions: (a)x is an optimal solution for (QP1) and strong duality holds; [14] it is shown that, for any feasible pointx, the condition ∇ x L(λ, γ ,x) = 0 is fulfilled with λ := (0, . . . , 0) and γ := (X Ax +Xa) and, moreover, for such λ and γ , is fulfilled and so is (a), by the previous part of the proof.
Conditions (11) and (15) in general are not fulfilled for a problem with bivalent constraints.

Example 3.11 Let
This also implies that C −x cl co T (C;x) so that Propositions 3.3 and 3.5 in general cannot be applied to problem (QP1).
Let us make some further comparison with the literature; until the end of this section we assume that f , g i , i = 1, . . . , m, are quadratic functions defined as in (5). According to Remark 3.7, the following results are all particular cases of Proposition 3.5.
where C := {x ∈ IR n : H x = d}, H is a ( p × n) matrix , and letx be feasible for (20).
The following assertions are equivalent: (a)x is an optimal solution and strong duality holds for (20); is positive semidefinite on Ker H. Consequently, when C := IR n , then (b) reduces to the following:

The Case with Two Quadratic Equality Constraints
In this section we analyse in details a quadratic problem with two quadratic equality constraints defined by where f , g i , i = 1, 2 are quadratic functions defined as in (5).
The standard Lagrangian associated with (21) L S : IR 2 × IR n −→ R is given by The following result is a consequence of Proposition 3.3.
Necessary or sufficient optimality conditions for a quadratic problem with two quadratic inequality constraints have been obtained in [1,18]. To the best of our knowledge, Theorem 4.1 is a new characterization of strong duality for a quadratic problem with two quadratic equality constraints.

Simultaneously Diagonalizable Quadratic Problems
In this section we characterize strong duality for a simultaneously diagonalizable quadratic problem with quadratic cone constraints, providing conditions that guarantee the existence of a convex reformulation. Our results generalize those obtained in [15] where two quadratic inequality constraints are considered under the assumption that the classic Slater condition is fulfilled. Consider problem (4) and assume that the matrices A and B i , i = 1, .., m are simultaneously diagonalizable, i.e. there exists an orthonormal matrix S order n, such that S AS = D 0 , S B i S = D i , S S = I , where D i are diagonal; we set D i = diag(γ i ), γ i := (γ i1 , . . . , γ in ) , i = 0, 1, . . . , m. We refer to [3] for an extensive description of the applications of this problem. Setting y = S x, then (4) can be written as follows: where P is a closed and convex cone in IR m ,g(y) := (g 1 (y), . . . ,g m (y)) andf ,g i : We assume that α = 0 and C = IR n . Now, set 1 2 y 2 i = z i , i = 1, . . . , n, then 1 2 y D i y = γ i z and (26) can be rewritten as follows: inf γ 0 z + a Sy :ĝ(y, z) ∈ −P, 1 2 Replacing the last n equality constraints with the corresponding inequalities, we obtain the following relaxation of (27) (and therefore of (26)): Let L : R m × R n → R be defined by L(λ, y) := 1 2 y D 0 y + a Sy + m i=1 λ igi (y) as the Lagrangian function associated with (26) and let sup λ∈P * inf y∈C L(λ, y) be the related dual problem. Similarly, let L R : R m × R n × R n × R n → R be defined by as the Lagrangian function associated with (28) and let be the corresponding dual problem. Moreover, if the supremum in the right-hand side of (29) is attained at (λ * , μ * ), then the supremum in the left-hand side is attained at λ * .
Proof Let us compute ψ(λ, μ) := inf y∈IR n z∈IR n L R (λ, μ, y, z). Note that By eliminating the variables μ j , we obtain: Therefore, and sup λ∈P * inf y∈IR n L(λ, y) = sup Notice that, if (31) does not hold, then sup λ∈P * inf y∈IR n L(λ, y) = −∞, which yields (29). The final assertion follows from (30). Assuming that τ R ∈ IR, following the image space approach introduced by Giannessi [10,11], we define the extended image associated with (28) by: It is possible to show that sincef andĝ are linear andĥ is convex, then E is a convex set, in fact F turns out to be a (IR + ×P × IR n + )-convex function. Many remarkable properties of a constrained extremum problem can be characterized (see [11]) by means of the set E, as in the next result. Proof It is known that condition (32) is equivalent to the fact that the duality gap is zero for (28) (see [17] Theorem 4.2, for a proof where it is assumed that the infimum τ R of (28) is attained, we notice that it is still valid if merely τ R ∈ IR). Then, by Proposition 5.1, the following relations hold: The proof is now straightforward.
Condition (32) is not easy to check: next result, based on a well-known constraints qualification, provides the connections with strong duality for (26).

Proposition 5.3
Assume that τ R ∈ IR and that the following condition holds for (28): Then, τ = τ R if and only if strong duality holds for (26).
Proof We first prove that (34) implies that strong duality holds for (28): to this aim we will apply Theorem 3.6 of [8] where (34) is requested as one of the assumptions. The other one is given by the following condition: where E is the extended image associated with (28). We now prove that (35) is fulfilled.
We have already observed that E is a convex set; we claim that Let us prove our claim. Notice that, since F is a continuous function then 0 ∈ cl E and since E is convex so is cl E, so that The reverse inclusion is obvious, so that cl co(E ∪ {0}) = cl E; by Theorem 6.3 of [19] we prove our claim. Now, since τ R ∈ IR, by Proposition 3.1 of [8] we have This proves that (35) is fulfilled and that strong duality holds for (28). Finally, Proposition 5.1 leads to the following relations: Assume that τ = τ R ; then the first inequality in (36) is fulfilled as equality and because of the second equality, the supremum is attained (see Proposition 5.1), i.e. strong duality holds for (26). Conversely, if strong duality holds for (26), then τ = max λ∈P * inf y∈IR n L(λ, y) and (36) yields τ = τ R .
We note that, when int P = ∅ the (34) collapses to the classic Slater condition.

Proposition 5.5 Assume
Proof We first note that, since (28) is a convex problem, then the KKT conditions guarantee the optimality of (ȳ,z) and (λ,μ,ȳ,z) is a saddle point of the Lagrangian function L R . Moreover, ifμ > 0, then the constraints 1 2 y 2 j − z j ≤ 0 are active for j = 1, .., n, which yields thatȳ is feasible for (27) and therefore for (26), which proves that τ = τ R andȳ is a global optimal solution for (26). By Proposition 5.1 the following relations hold: where the last two equalities follow from the fact that (λ,μ,ȳ,z) is a saddle point of where the last equality is due to Proposition 5.1, which proves that strong duality holds for (26).
We provide a sufficient condition for (34) to be fulfilled.
In the particular case where the feasible set of (28) is defined by explicit equality and inequality constraints, i.e. P := {0} s × IR m−s + , for 0 ≤ s ≤ m, we obtain a refinement of Proposition 5.3. .

Remark 5.8
Computing explicitly the gradients ofĝ andĥ, then (i) and (ii) of Proposition 5.7 can be written as We note that in [15] the Slater-type condition (34) has been considered as a blanket assumption. Finally, we provide a refinement of Corollary 5.4.

Conclusions
We have considered a quadratic programming problem with general quadratic cone constraints and an additional geometric constraint. We have established necessary and sufficient conditions for global optimality for a KKT point or in the presence of the property of strong duality, considering in details the case where the feasible set is defined by two quadratic equality constraints. As a further application, we have obtained conditions that guarantee the existence of a convex reformulation of a simultaneous diagonalizable quadratic problem.