A Note on the McCormick Second-Order Constraint Qualification

FAZZIO, N. S.; NCHEZ, M. D. SÁ; SCHUVERDT, M. L.

doi:10.5540/tcam.2022.023.04.00769

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Articles • Trends Comput. Appl. Math. 23 (4) • Oct-Dec 2022 • https://doi.org/10.5540/tcam.2022.023.04.00769 linkcopy

A Note on the McCormick Second-Order Constraint Qualification

Authorship SCIMAGO INSTITUTIONS RANKINGS

The study of optimality conditions and constraint qualification is a key topic in nonlinear optimization. In this work, we present a reformulation of the well-known second-order constraint qualification described by McCormick in ¹⁷. This reformulation is based on the use of feasible arcs, but is independent of Lagrange multipliers. Using such a reformulation, we can show that a local minimizer verifies the strong second-order necessary optimality condition. We can also prove that the reformulation is weaker than the known relaxed constant rank constraint qualification in ¹⁹. Furthermore, we demonstrate that the condition is neither related to the MFCQ + WCR in ⁸ nor to the CCP2 condition, the companion constraint qualification associated with the second-order sequential optimality condition AKKT 2 in ⁵.

Keywords:
Nonlinear programming; second-order optimality conditions; constraint qualification

Authorship

person N. S. FAZZIO

school CONICET, Department of Mathematics, FCE, University of La Plata, 1900 La Plata Bs. As., Argentina - E-mail: nfazzio@mate.unlp.edu.arUniversity of La PlataArgentinaLa Plata, Bs. As., Argentina CONICET, Department of Mathematics, FCE, University of La Plata, 1900 La Plata Bs. As., Argentina - E-mail: nfazzio@mate.unlp.edu.ar

person M. D. SÁ NCHEZ

schoolDepartment of Mathematics, FCE, University of La Plata, 1900 La Plata Bs. As., Argentina - E-mail: dsanchez@mate.unlp.edu.arUniversity of La PlataArgentinaLa Plata, Bs. As., ArgentinaDepartment of Mathematics, FCE, University of La Plata, 1900 La Plata Bs. As., Argentina - E-mail: dsanchez@mate.unlp.edu.ar

person M. L. SCHUVERDT *

schoolCONICET, Department of Mathematics, FCE, University of La Plata, 1900 La Plata Bs. As., Argentina - E-mail: schuverd@mate.unlp.edu.ar University of La PlataArgentinaLa Plata, Bs. As., ArgentinaCONICET, Department of Mathematics, FCE, University of La Plata, 1900 La Plata Bs. As., Argentina - E-mail: schuverd@mate.unlp.edu.ar

0000-0002-3296-7895

person*Corresponding author: María Laura Schuverdt - E-mail:
schuverd@mate.unlp.edu.ar

SCIMAGO INSTITUTIONS RANKINGS

CONICET, Department of Mathematics, FCE, University of La Plata, 1900 La Plata Bs. As., Argentina - E-mail: nfazzio@mate.unlp.edu.arUniversity of La PlataArgentinaLa Plata, Bs. As., Argentina CONICET, Department of Mathematics, FCE, University of La Plata, 1900 La Plata Bs. As., Argentina - E-mail: nfazzio@mate.unlp.edu.ar

Department of Mathematics, FCE, University of La Plata, 1900 La Plata Bs. As., Argentina - E-mail: dsanchez@mate.unlp.edu.arUniversity of La PlataArgentinaLa Plata, Bs. As., ArgentinaDepartment of Mathematics, FCE, University of La Plata, 1900 La Plata Bs. As., Argentina - E-mail: dsanchez@mate.unlp.edu.ar

CONICET, Department of Mathematics, FCE, University of La Plata, 1900 La Plata Bs. As., Argentina - E-mail: schuverd@mate.unlp.edu.ar University of La PlataArgentinaLa Plata, Bs. As., ArgentinaCONICET, Department of Mathematics, FCE, University of La Plata, 1900 La Plata Bs. As., Argentina - E-mail: schuverd@mate.unlp.edu.ar

Figures | Formulas

Figures (1)
Formulas (46)

Thumbnail

Figure 1: Relationship of second-order CQs. An arrow between two CQs means that one is strictly stronger than the other.

(1.1)

(1.2)

(1.3)

(1.4)

(1.5)

(1.6)

(2.1)

(3.1)

(3.2)

(3.3)

imageFigure 1: Relationship of second-order CQs. An arrow between two CQs means that one is strictly stronger than the other. open_in_new

(1.1)

Minimize f (x) s.t. h_{i} (x) = 0, for i = 1, \dots, m g_{j} (x) \leq 0, for j = 1, \dots, p

Ω = {x \in ℝ^{n} : h (x) = 0, g (x) \leq 0}

μ_{0} \nabla f (x) + \sum_{i = 1}^{m} λ_{i} \nabla h_{i} (x) + \sum_{j = 1}^{p} μ_{j} \nabla g_{j} (x) = 0, μ_{j} \geq 0, μ_{j} g_{j} (x) = 0, j = 1, \dots, p .

(1.2)

\nabla f (x) + \sum_{i = 1}^{m} λ_{i} \nabla h_{i} (x) + \sum_{j = 1}^{p} μ_{j} \nabla g_{j} (x) = 0, μ_{j} \geq 0, μ_{j} g_{j} (x) = 0, j = 1, \dots, p .

(1.3)

l (x, λ, μ) = f (x) + \sum_{i = 1}^{m} λ_{i} h_{i} (x) + \sum_{j = 1}^{p} μ_{j} g_{j} (x) .

C (x) = \{d \in ℝ^{n} : \begin{matrix} \nabla h_{i} (x)^{⊺} d = 0, & i = 1, \dots, m \\ \nabla g_{j} (x)^{⊺} d = 0, & j \in A (x) : μ_{j} > 0 \\ \nabla g_{j} (x)^{⊺} d \leq 0, & j \in A (x) : μ_{j} = 0 \end{matrix}\}

(1.4)

d^{T} \nabla^{2} l (x, λ, μ) d \geq 0,

C_{0} (x) = \{d \in ℝ^{n} : \begin{matrix} \nabla h_{i} (x)^{⊺} d = 0, & i = 1, \dots, m \\ \nabla g_{j} (x)^{⊺} d = 0, & j \in A (x) \end{matrix}\} .

(1.5)

d^{T} \nabla^{2} l (x, λ, μ) d \geq 0,

(1.6)

L (x) = {d \in ℝ^{n} : \nabla h_{i} (x)^{⊺} d = 0, i = 1, \dots, m; \nabla g_{j} (x)^{⊺} d \leq 0, j \in A (x)}

\begin{matrix} h_{i} (α (t)) = 0, i = 1 \dots, m; \\ g_{j} (α (t)) = 0, \forall j \in A (x) . \end{matrix}

(2.1)

\begin{array}{l} h_{i} (α (t)) = 0, i = 1, \dots, m; \\ g_{j} (α (t)) = 0, \forall j \in A (x) : \nabla g_{j} (x)^{⊺} d = 0; \\ g_{j} (α (t)) < 0, \forall j \in A (x) : \nabla g_{j} (x)^{⊺} d < 0 . \end{array}

- \nabla f (x^{*}) \in (L (x^{*})) ° = \{z \in ℝ^{n} : z = \sum_{i = 1}^{m} λ_{i} \nabla h_{i} (x^{*}) + \sum_{j = 1}^{p} μ_{j} \nabla g_{j} (x^{*}), μ_{j} \geq 0 \forall j \in A (x^{*})\} .

ℓ' (0) = \nabla_{x} l (x^{*}, λ^{*}, μ^{*})^{T} d = 0

ℓ'' (0) = d^{T} \nabla_{x}^{2} l (x^{*}, λ^{*}, μ^{*}) d .

ℓ (t) = f (x^{*}) + \frac{t^{2}}{2} d^{T} \nabla_{x}^{2} l (x^{*}, λ^{*}, μ^{*}) d + o (t^{2}) .

f (α (t)) = f (x^{*}) + \frac{t^{2}}{2} d^{T} \nabla_{x}^{2} l (x^{*}, λ^{*}, μ^{*}) d + o (t^{2}) .

0 \leq f (α (t)) - f (x^{*}) \leq \frac{t^{2}}{2} d^{T} \nabla_{x}^{2} l (x^{*}, λ^{*}, μ^{*}) d + o (t^{2}) .

{\nabla h_{i} (y)}_{i = 1, \dots, m} \cup {\nabla g_{j} (y)}_{j \in J}

(3.1)

C (y, t) : = c (x^{*} + t d + y) .

C (0, 0) = c (x^{*}) = 0

C_{y} (y, t) = J c (x^{*} + t d + y)

C_{y} (0, 0) = J c (x^{*}) .

(3.2)

C ((r (t), 0), t) = 0

(3.3)

C_{y^{1}} (0, 0) r' (0) + C_{t} (0, 0) = 0 .

C_{t} (0, 0) = J c (x^{*}) d = 0 .

α (t) = x^{*} + t d + r (t) .

c (α (t)) = c (x^{*} + t d + r (t)) = C ((r (t), 0), t) = 0

ϕ (0) = c_{i} (α (0)) = c_{i} (x^{*}) = 0

ϕ' (0) = \nabla c_{i} (x^{*})^{⊺} α' (0) = \nabla c_{i} (x^{*})^{⊺} d < 0 .

g_{1} (x_{1}, x_{2}) = x_{1}; g_{2} (x_{1}, x_{2}) = x_{1} e^{x_{2}} .

\nabla g_{1} (x_{1}, x_{2}) = (1, 0), \nabla g_{1} (0, 0) = (1, 0); \nabla g_{2} (x_{1}, x_{2}) = (e^{x_{2}}, x_{1} e^{x_{2}}), \nabla g_{2} (0, 0) = (1, 0) .

{\nabla h_{i} (y)}_{i = 1, \dots, m} \cup {\nabla g_{j} (y)}_{j \in A (x)}

g_{1} (x_{1}, x_{2}) = x_{1}; g_{2} (x_{1}, x_{2}) = x_{1} - x_{2}^{2}; g_{3} (x_{1}, x_{2}) = x_{1} + x_{2}; g_{4} (x_{1}, x_{2}) = - x_{1} - x_{2} .

\nabla g_{1} (x_{1}, x_{2}) = (1, 0); \nabla g_{2} (x_{1}, x_{2}) = (1, - 2 x_{2}); \nabla g_{3} (x_{1}, x_{2}) = (1, 1); \nabla g_{4} (x_{1}, x_{2}) = (- 1, - 1);

\begin{array}{lr} g_{1} (α (t)) = t d_{1} < 0, & \nabla g_{1} (0, 0)^{⊺} d < 0; \\ g_{2} (α (t)) = t d_{1} (1 - t d_{1}) < 0, & \nabla g_{2} (0, 0)^{⊺} d < 0; \\ g_{3} (α (t)) = 0, & \nabla g_{3} (0, 0)^{⊺} d = 0; \\ g_{4} (α (t)) = 0, & \nabla g_{4} (0, 0)^{⊺} d = 0 . \end{array}

g_{1} (x_{1}, x_{2}, x_{3}) = - x_{1} + x_{2} - x_{3}^{2}; g_{2} (x_{1}, x_{2}, x_{3}) = - x_{1} - x_{2}; g_{3} (x_{1}, x_{2}, x_{3}) = - x_{1}; g_{4} (x_{1}, x_{2}, x_{3}) = - x_{1}^{2} - x_{2}^{2} - x_{3}^{2} + x_{3} .

\begin{array}{l} \nabla g_{1} (x_{1}, x_{2}, x_{3}) = (- 1, 1, - 2 x_{3}), & \nabla g_{1} (0, 0, 0) = (- 1, 1, 0); \\ \nabla g_{2} (x_{1}, x_{2}, x_{3}) = (- 1, - 1, 0), & \nabla g_{2} (0, 0, 0) = (- 1, - 1, 0); \\ \nabla g_{3} (x_{1}, x_{2}, x_{3}) = (- 1, 0, 0), & \nabla g_{3} (0, 0, 0) = (- 1, 0, 0); \\ \nabla g_{4} (x_{1}, x_{2}, x_{3}) = (- 2 x_{1}, - 2 x_{2}, - 2 x_{3} + 1), & \nabla g_{4} (0, 0, 0) = (0, 0, 1) . \end{array}

C_{W} (x, x^{*}) = \{d \in ℝ^{n} : \nabla h_{i} (x)^{⊺} d = 0, i = 1, \dots, m; \nabla g_{j} (x)^{⊺} d = 0, j \in A (x^{*})\} .

\underset{}{\cup} \{\begin{matrix} (\sum_{i = 1}^{m} λ_{i} \nabla h_{i} (x) + \sum_{j \in A (x^{*})} μ_{j} \nabla g_{j} (x), H), \\ such that H ⪯ \sum_{i = 1}^{m} λ_{i} \nabla^{2} h_{i} (x) + \sum_{j \in A (x^{*})} μ_{j} \nabla^{2} g_{j} (x) on C_{W} (x, x^{*}) \end{matrix}\}

\underset{x \to x^{*}}{\lim \sup} K_{2}^{W} (x) \subset K_{2}^{W} (x^{*}) .

g_{1} (x_{1}, x_{2}) = - x_{1}; g_{2} (x_{1}, x_{2}) = x_{1} + \max {x_{1}, 0}^{2} e^{x_{2}^{2}} .

\nabla g_{1} (x_{1}, x_{2}) = (- 1, 0), \nabla g_{2} (x_{1}, x_{2}) = (1 + 2 \max {x_{1}, 0} e^{x_{2}^{2}}, 2 x_{2} \max {0, x_{1}}^{2} e^{x_{2}^{2}}), \nabla g_{2} (0, 0) = (1, 0);

\begin{array}{lr} g_{1} (α (t)) = 0, & \nabla g_{1} (0, 0)^{⊺} d = 0; \\ g_{2} (α (t)) = 0, & \nabla g_{2} (0, 0)^{⊺} d = 0 . \end{array}

h_{1} (x_{1}, x_{2}) = x_{1}; g_{1} (x_{1}, x_{2}) = - x_{1}^{2} + x_{2}; g_{2} (x_{1}, x_{2}) = - x_{1}^{2} + x_{2}^{3} .

\begin{array}{l} \nabla h_{1} (x_{1}, x_{2}) = (1, 0), & \nabla h_{1} (0, 0) = (1, 0); \\ \nabla g_{1} (x_{1}, x_{2}) = (- 2 x_{1}, 1), & \nabla g_{1} (0, 0) = (0, 1); \\ \nabla g_{2} (x_{1}, x_{2}) = (- 2 x_{1}, 3 x_{2}^{2}), & \nabla g_{2} (0, 0) = (0, 0) . \end{array}

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