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The linearized quadratic subproblem in sequential quadratic programming

Definition

Use the $C^{2}$ nonlinear program convention of Inequality constrained programs, tangent cones, constraint qualifications, and KKT: minimize a $C^{2}$ objective $f : U \to R$ on an open $U \subseteq R^{n}$ , subject to $h (x) = 0$ and componentwise $g (x) \leq 0$ . Fix a current point $x \in U$ and a symmetric positive-definite matrix $B \in R^{n \times n}$ . The SQP quadratic subproblem is $\min_{p \in R^{n}} \nabla f (x)^{T} p + \frac{1}{2} p^{T} B p subject to h (x) + D h (x) p = 0, g (x) + D g (x) p \leq 0.$ The variable $p$ is the proposed displacement. Its feasible set is the polyhedron defined by the displayed affine constraints. A constraint index $i$ is active at $p$ when $g_{i} (x) + D g_{i} (x) p = 0$ . For equality multiplier $ν$ and inequality multipliers $λ \geq 0$ , a KKT tuple for the subproblem satisfies primal feasibility, $B p + \nabla f (x) + D h (x)^{T} ν + D g (x)^{T} λ = 0, λ_{i} (g_{i} (x) + D g_{i} (x) p) = 0 (1 \leq i \leq m) .$ Rows of $D h (x)$ and active rows of $D g (x)$ are stacked in that order in the subproblem KKT matrix $[\begin{array}{cc} B & A^{T} \\ A & 0 \end{array}]$ .

All inequality signs and multiplier signs here agree with the source program: $g_{i} \leq 0$ and $λ_{i} \geq 0$ .

Facts & Assumptions

Given: A $C^{2}$ program, a current point $x$ , and a symmetric positive- definite model matrix $B$ .

[F1]

Inequality constrained programs, tangent cones, constraint qualifications, and KKT fixes the program's equality and componentwise $g_{i} \leq 0$ inequality convention, with Lagrangian $f + ν^{T} h + λ^{T} g$ and $λ \geq 0$ .

[A1]

The model matrix $B$ is symmetric positive definite.

Proof

1.1givenA1F1algebra

Differentiability at $x$ gives the affine first-order models $h (x) + D h (x) p$ and $g (x) + D g (x) p$ ; the positive-definite quadratic model is finite for every $p \in R^{n}$ . Thus the displayed feasible set is a well-defined polyhedron, possibly empty.

2.1F1algebra∎

For an active affine inequality, the multiplier convention in the Lagrangian contributes $D g_{i} (x)^{T} λ_{i}$ to stationarity, while complementarity is $λ_{i} (g_{i} (x) + D g_{i} (x) p) = 0$ . Hence the stated KKT tuple uses the same sign convention as [F1].

Depends on

Inequality constrained programs, tangent cones, constraint qualifications, and KKT

Used by

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Sources

Jorge Nocedal and Stephen J. Wright, Numerical Optimization, 2nd ed. (standard reference, not scraped)
Ralf Herzog, Algorithms and Preconditioning in PDE-Constrained Optimization, lecture notes (standard reference, not scraped)