x+y ≤ b₁; 2x+y ≤ 5; x,y ≥ 0. Only b₁ changes.
Teal: resource 1 · Graphite: resource 2 · Sand: feasible region · Amber point: optimum
Teal: actual optimum · Dashed amber: old-price prediction · Graphite point: current value
f(u,v)=(u−1)²+4(v+2)²; ∇f=(2(u−1),8(v+2)). Minimum at (1,−2).
Accept if f(r+αp) ≤ f(r)+c₁αgᵀp. Here p=−g.
| Iteration | Trial α | Actual f | Armijo target | Decision |
|---|
q(x)=(x²−1)²; g=4x(x²−1); H=12x²−4.
Predict each value before clicking. In Backtracking, use c₁=0.1 and ρ=0.5: reject 1, 0.5 and 0.25; accept 0.125. After moving, repeat the search with the new gradient. The next accepted step is 0.5 and reaches the minimum for this particular example.
Reference: Boyd and Vandenberghe, Convex Optimization, Chapter 9. All examples here are calculated directly. No data is submitted by this page.