In Business Analytics and Linear Programming & Optimization, modern solvers like HiGHS report “dual values” for each constraint. These dual values are exactly the shadow prices economists and operations researchers use to measure the value of relaxing a constraint.
This page explains what HiGHS dual values mean, why they equal shadow prices, and how to interpret them for binding and non‑binding constraints.
What Are HiGHS Dual Values?
For a maximization problem, a dual value tells you:
“If I increase this constraint’s RHS by 1, the objective increases by X.”
For minimization, the interpretation is reversed.
Why HiGHS Dual Values Equal Shadow Prices
1. Dual values come from the LP dual problem
Every LP has a dual. The dual variables correspond to constraints in the primal.
2. Shadow prices measure marginal value of resources
A shadow price tells you how much the objective improves when you relax a constraint.
3. HiGHS solves both primal and dual simultaneously
Modern simplex and interior‑point methods compute dual variables as part of the solution.
4. Complementary slackness links binding constraints to positive duals
If a constraint is binding, its dual value equals the shadow price. If it is slack, its shadow price is zero.
5. Dual values are the gradient of the optimal value function
Mathematically, the dual value is the derivative of the optimal objective with respect to the RHS.
How to Interpret HiGHS Dual Values (Step by Step)
Step 1: Identify whether the constraint is binding
If the constraint is tight at the optimum, its dual value may be nonzero. If it has slack, the dual value must be zero.
Step 2: Check the sign convention
- Maximization → dual values are nonnegative for ≤ constraints.
- Minimization → dual values are nonpositive for ≥ constraints.
Step 3: Interpret the magnitude
A dual value of 7 means:
“Relaxing this constraint by 1 unit improves the objective by 7.”
Step 4: Understand the economic meaning
Dual values measure the marginal value of scarce resources:
- machine hours
- labor availability
- budget limits
- material constraints
Step 5: Use dual values for sensitivity analysis
HiGHS dual values are valid only within the allowable RHS range where the basis remains optimal.
Numerical Example
Consider the LP:
\[ \text{Max } z = 5x_1 + 4x_2 \]
\[ \begin{aligned} 2x_1 + x_2 &\le 10 \\ x_1 + 3x_2 &\le 12 \\ x_1, x_2 &\ge 0 \end{aligned} \]
Suppose HiGHS returns dual values:
- Constraint 1: 1.5
- Constraint 2: 0
Interpretation
- Constraint 1 is binding → relaxing it by 1 increases z by 1.5.
- Constraint 2 has slack → relaxing it has no effect.
Thus, the shadow price of constraint 1 is 1.5.
Common Mistakes
- Thinking dual values apply outside the allowable RHS range.
- Misinterpreting signs for minimization problems.
- Assuming slack constraints can have nonzero duals.
- Confusing reduced costs with dual values.
- Ignoring complementary slackness.
Why This Matters
Understanding HiGHS dual values helps you:
- perform sensitivity analysis
- value scarce resources
- interpret binding vs non‑binding constraints
- understand duality theory
- optimize real‑world systems more effectively
Shadow prices are essential for managerial decision‑making and optimization.
Related Topics
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