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Powell damping repairs a negative-curvature trial pair

Example

Consider the nonconvex function f(x1,x2)=−12x12+12x22 at x=(1,0)T. The gradient is g=(−1,0)T. The trial direction p=(1,0)T and trial step α=1 give s=(1,0)T and y=(−1,0)T. Use B=H=I to compare the raw and Powell-damped updates.

Facts & Assumptions

Given: The function, point, trial direction and step, and initial matrices in the Example.

[F1]

Bfgs curvature pairs and Hessian/inverse updates: an undamped BFGS update requires sTy>0; Powell damping uses c=sTBs and, if sTy<0.2c, forms y~=θy+(1−θ)Bs with θ=0.8c/(c−sTy).

[F2]

Powell damping gives a strictly positive curvature pair: this damping rule gives sTy~=0.2c>0 in the damped branch.

[F3]

Bfgs updates preserve the secant equations, symmetry, and positive definiteness: for SPD B and positive damped curvature, the BFGS Hessian update remains SPD and satisfies B+s=y~.

[F4]

Full-memory Bfgs with Wolfe search, descent reset, and Powell damping: the normal search branch requires a strong Wolfe step and reports line-search failure if its finite search does not return one.

Verification

Given: The function, point, trial direction and step, and initial matrices in the Example.

[F1] Bfgs curvature pairs and Hessian/inverse updates: an undamped BFGS update requires sTy>0; Powell damping uses c=sTBs and, if sTy<0.2c, forms y~=θy+(1−θ)Bs with θ=0.8c/(c−sTy).

[F2] Powell damping gives a strictly positive curvature pair: this damping rule gives sTy~=0.2c>0 in the damped branch.

[F3] Bfgs updates preserve the secant equations, symmetry, and positive definiteness: for SPD B and positive damped curvature, the BFGS Hessian update remains SPD and satisfies B+s=y~.

[F4] Full-memory Bfgs with Wolfe search, descent reset, and Powell damping: the normal search branch requires a strong Wolfe step and reports line-search failure if its finite search does not return one.

1.1givenF4algebra

Here gTp=−1<0, and f(x+p)−f(x)=−3/2. The Armijo inequality holds for c1=10−4, but the directional derivatives are ϕ′(0)=−1 and ϕ′(1)=−2. For c2=2/5, the Wolfe curvature test would require −2≥−2/5, which is false; the strong curvature test fails as well. Thus this is a nonconvex trial pair, not a step accepted by the strong-Wolfe search of [F4].

1.2givenF1algebra

The raw pair has sTy=−1, so its BFGS curvature denominator is negative. Direct substitution into the formal Hessian-update expression gives I−ssT+yyT/(−1)=diag⁡(−1,1), whose quadratic form at e1=(1,0)T is −1; it is not SPD.

1.3givenF1F2algebra

Here c=sTBs=1 and sTy=−1<0.2c, so θ=0.8/(1−(−1))=0.4 and y~=0.4(−1,0)T+0.6(1,0)T=(0.2,0)T. Therefore sTy~=0.2>0, as also follows from [F2].

2.1givenF1F3step 1.3algebra∎

The damped update is B+=I−ssT+y~y~TsTy~=diag⁡(0.2,1). It is SPD and B+s=(0.2,0)T=y~, in agreement with [F3]. The example shows the algebraic role of damping; because the trial fails Wolfe, the normal line-search algorithm rejects it rather than treating it as a successful BFGS iteration.

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