Track when a shadow price remains valid

Try

Maximize 3x+2y subject to x+y≤b, x≤2 and x,y≥0, where the resource amount b≥0 can change. When b is small, assign all resource to x, the higher-value activity. Once x reaches its cap, assign further resource to y. This produces a piecewise-linear optimal value. Sensitivity analysis should identify these regime changes, not report one multiplier as an eternal value for every extra unit of resource.

Predict

Predict the marginal value before and after b=2. Before the cap binds, an extra unit creates three units of objective; afterward it creates two. At the breakpoint there is no single two-sided derivative. A dual optimal price still provides a supporting bound, but several prices can be valid there. The distinction between a supporting price and a derivative becomes essential when the optimal basis is not unique or changes under perturbation.

Experiment

Derive both feasible optimizers and upper bounds for the two regimes. For b≤2 use 3x+2y≤3(x+y)≤3b. For b≥2 use 3x+2y=2(x+y)+x≤2b+2.

Resource b	One optimizer	Value v(b)	Regime slope
1	(1,0)	3	3
2	(2,0)	6	Left 3, right 2
4	(2,2)	10	2

Check that the two formulas agree at the breakpoint. Then use the price three from the first regime to predict the value at b=4. It gives the upper bound twelve, not the actual value ten. The bound remains valid while exact linear prediction fails outside its regime.

Understand

For max cᵀx with Ax≤b,x≥0, fix A and c and let y⋆ be a dual optimal solution at b. Dual constraints do not depend on b, so y⋆ remains dual feasible after replacing b by b+Δ. Weak duality gives v(b+Δ)≤v(b)+y⋆ᵀΔ whenever the new primal is feasible. This is a global supporting upper bound, not automatically equality. Viewed as a function of b over its feasible parameter domain, the value is concave because it is the infimum of dual affine bounds when strong duality applies.

For a fixed optimal equality-form basis B, changing only b leaves reduced costs unchanged. The basis remains optimal as long as B⁻¹(b+Δ)≥0. Within that basis-feasibility region, the value is exactly c_BᵀB⁻¹(b+Δ), and its slope is the basis dual vector. Changing objective coefficients instead affects reduced costs, while changing A can affect both the basis inverse and feasibility. A sensitivity statement must identify which input is varying and which quantities are held fixed.

Apply

At b=2, the dual of the worked model minimizes bu+2v with u+v≥3, u≥2 and u,v≥0. Every pair with u between two and three and v=3−u has objective six. Derive this segment of optimal prices. For a positive perturbation, u=2 gives the exact slope; for a negative perturbation, u=3 does. Intermediate prices remain supporting bounds but do not equal either one-sided derivative. Multiple optimal multipliers therefore have a concrete economic and geometric meaning.

Use the primal–dual experiment to inspect a different LP’s breakpoints. Keep its coefficients separate from this worked example. Numerical agreement in one regime is not evidence that the same slope survives a basis change, an integrality restriction or a changed objective.

Continue

Report a shadow price with its parameter convention, validity region and any nonuniqueness. If the underlying coefficients are uncertain, distinguish mathematical sensitivity from an empirical prediction. Continue to Gradient descent for nonlinear iterative methods. Their progress certificates concern smoothness, curvature and step sizes rather than LP bases, but the discipline of stating the conditions behind a numerical claim remains the same.