MIP Modeling Expert

When to Use This Skill

Use this skill when you need help with:

• Modeling discontinuous variables (variables that must be 0 or within specific bounds)
• Handling fixed costs or setup costs in optimization models
• Formulating either-or constraints (at least one of two constraints must hold)
• Conditional constraints (if constraint A holds, then constraint B must hold)
• Piecewise linear approximations of nonlinear functions
• Special Ordered Sets (SOS1, SOS2) for efficient branch-and-bound
• Linearizing products of variables (binary × binary, binary × continuous, continuous × continuous)
• Converting nonlinear models to MIP formulations

Overview

This skill provides expert guidance on Integer Linear Programming Tricks based on the AIMMS Modeling Guide. These techniques transform complex, nonlinear, or logical requirements into linear integer programming formulations that can be solved by modern MIP solvers (CPLEX, Gurobi, etc.).

Core Principle: Many practical optimization problems cannot be formulated as pure linear programs because they involve:

• Discontinuities (jumps in cost functions or variable bounds)
• Logical decisions (either-or, if-then relationships)
• Nonlinear terms (products of variables, quadratic functions)

These challenges can be addressed using indicator variables (binary 0-1 variables) and Big-M formulations.

Key Techniques

1. Indicator Variables

Binary indicator variables are the foundation of most MIP tricks. An indicator variable y ∈ {0,1} signals a specific state:

y = 0  →  State A
y = 1  →  State B

2. Discontinuous Variables

Problem: A variable must be either 0 OR within bounds [l, u]:

x = 0  or  l ≤ x ≤ u

Solution: Introduce binary indicator y:

x ≤ uy
x ≥ ly
y binary

When y=0: x=0. When y=1: l ≤ x ≤ u.

Applications:

• Batch size restrictions (suppliers require minimum order quantities)
• Setup costs in manufacturing

3. Fixed Costs

Problem: Cost function with a jump discontinuity:

C(x) = {  0        if x = 0
       {  k + cx   if x > 0

where k is the fixed cost and c is the variable cost.

Solution: Introduce binary y and upper bound u:

Minimize:  ky + cx

Subject to:
    x ≤ uy
    x ≥ 0
    y binary

When y=0: forces x=0, cost is 0. When y=1: allows x>0, cost is k + cx.

Applications:

• Equipment setup costs
• Opening a facility or warehouse

4. Either-Or Constraints

Problem: At least one of two constraints must hold:

Constraint (1): Σ a₁ⱼxⱼ ≤ b₁   OR
Constraint (2): Σ a₂ⱼxⱼ ≤ b₂

Solution: Use binary y and Big-M constants M₁, M₂:

Σ a₁ⱼxⱼ ≤ b₁ + M₁y
Σ a₂ⱼxⱼ ≤ b₂ + M₂(1-y)
y binary

When y=0: Constraint (1) is enforced, (2) is relaxed When y=1: Constraint (2) is enforced, (1) is relaxed

Critical: Choose M values as tight as possible while ensuring they don't restrict feasible solutions.

Applications:

• Alternative production modes
• Route selection

5. Conditional Constraints (If-Then)

Problem: If constraint (1) holds, then constraint (2) must hold:

If  Σ a₁ⱼxⱼ ≤ b₁  then  Σ a₂ⱼxⱼ ≤ b₂

Logical Equivalence:

(A → B)  ≡  (¬A ∨ B)

This translates to: "Either constraint (1) is violated OR constraint (2) holds"

Solution: Use binary y, Big-M M, lower bound L, and small tolerance ε:

Σ a₁ⱼxⱼ ≥ b₁ + ε - Ly
Σ a₂ⱼxⱼ ≤ b₂ + M(1-y)
y binary

Applications:

• Safety constraints that activate when production exceeds thresholds
• Regulatory requirements

6. Special Ordered Sets (SOS)

SOS Type 1 (SOS1)

Definition: At most ONE variable in the set can be nonzero.

Σᵢ yᵢ ≤ 1   (where yᵢ are 0-1 variables)

More generally, for continuous variables 0 ≤ xᵢ ≤ uᵢ:

Σᵢ aᵢxᵢ ≤ b   with at most one xᵢ nonzero

Performance: When variables have a natural ordering, solvers use specialized branching strategies that reduce the search tree.

Example: Warehouse size must be exactly one of: {10000, 20000, 40000, 50000} sq ft.

x₁ + x₂ + x₃ + x₄ = 1  (SOS1)
size = 10000x₁ + 20000x₂ + 40000x₃ + 50000x₄

SOS Type 2 (SOS2)

Definition: At most TWO variables can be nonzero, and they must be adjacent in the ordering.

Used extensively in piecewise linear approximations (see below).

7. Piecewise Linear Approximations

Problem: Approximate a nonlinear function f(x) with a piecewise linear function.

Requirements:

• Function must be separable: F(x₁, x₂, ...) = f₁(x₁) + f₂(x₂) + ...
• Define breakpoints: x₁, x₂, x₃, ..., xₙ

λ-Formulation:

For breakpoints x₁, x₂, x₃, x₄ with function values f(x₁), f(x₂), f(x₃), f(x₄):

x = λ₁x₁ + λ₂x₂ + λ₃x₃ + λ₄x₄
f̃(x) = λ₁f(x₁) + λ₂f(x₂) + λ₃f(x₃) + λ₄f(x₄)
λ₁ + λ₂ + λ₃ + λ₄ = 1
λᵢ ≥ 0

Plus the adjacency requirement: At most two adjacent λᵢ can be nonzero.

This is exactly a SOS2 constraint.

When Adjacency is Redundant:

• Minimizing convex functions (slopes are non-decreasing)
• Maximizing concave functions (slopes are non-increasing)

In these cases, the LP relaxation automatically satisfies adjacency, and no MIP solve is needed!

When MIP is Required:

• Non-convex functions (adjacency must be enforced with SOS2)

Example: Approximate f(x) = ½x² on [0,4] with breakpoints at 0, 1, 2, 4:

Breakpoints:     x:  0   1   2   4
Function values: f:  0  0.5  2   8

x = λ₁·0 + λ₂·1 + λ₃·2 + λ₄·4
f̃ = λ₁·0 + λ₂·0.5 + λ₃·2 + λ₄·8
λ₁ + λ₂ + λ₃ + λ₄ = 1  (SOS2)

8. Eliminating Products of Variables

Product of Two Binary Variables

Problem: Linearize y = x₁x₂ where x₁, x₂ ∈ {0,1}

Solution:

y ≤ x₁
y ≤ x₂
y ≥ x₁ + x₂ - 1
y binary

Product of Binary and Continuous

Problem: Linearize y = x₁x₂ where x₁ ∈ {0,1} and 0 ≤ x₂ ≤ u

Solution:

y ≤ ux₁
y ≤ x₂
y ≥ x₂ - u(1 - x₁)
y ≥ 0

Verification:

• If x₁=0: constraints force y=0
• If x₁=1: constraints force y=x₂

Product of Two Continuous Variables

Problem: Linearize x₁x₂ where both are continuous

Solution: Convert to separable form using:

y₁ = ½(x₁ + x₂)
y₂ = ½(x₁ - x₂)

Then: x₁x₂ = y₁² - y₂²

Now approximate y₁² and y₂² using piecewise linear functions.

Special Case: If l₁, l₂ ≥ 0 and x₁ appears ONLY in products, substitute z = x₁x₂ and add:

l₁x₂ ≤ z ≤ u₁x₂

Quick Reference Patterns

Pattern 1: Binary Indicator Linking

Variable x must satisfy:  x = 0  OR  l ≤ x ≤ u

Formulation:
    x ≤ uy
    x ≥ ly
    y ∈ {0,1}

Pattern 2: Big-M Constraint Activation

Binary y controls whether constraint is active:

y=0: Constraint enforced
y=1: Constraint relaxed

Formulation:
    Σ aⱼxⱼ ≤ b + My

Pattern 3: SOS2 Piecewise Linear

For convex minimization or concave maximization:
    Just use λ-formulation (adjacency is automatic)

For non-convex:
    Mark the convexity constraint as SOS2 property

Pattern 4: Product Linearization

Binary × Binary:      3 constraints + 1 binary variable
Binary × Continuous:  4 constraints + 1 continuous variable
Continuous × Continuous: Convert to separable, then use piecewise linear

Common Applications

1. Production Planning

   • Setup costs (fixed cost trick)
   • Batch size restrictions (discontinuous variables)
   • Minimum production runs

2. Logistics

   • Facility location with fixed opening costs
   • Route selection (either-or constraints)
   • Vehicle capacity restrictions

3. Scheduling

   • Machine setup times
   • Conditional precedence constraints
   • Time windows

4. Finance

   • Portfolio optimization with transaction costs
   • Quantity discounts (piecewise linear)
   • Conditional investments

5. Energy Systems

   • Unit commitment (generators on/off with startup costs)
   • Alternative fuel sources (either-or)
   • Nonlinear efficiency curves

Best Practices

1. Choose Big-M Values Carefully

   • Too large: numerical instability, weak LP relaxation
   • Too small: may cut off feasible solutions
   • Best: Derive from problem-specific bounds

2. Use SOS When Natural Ordering Exists

   • Dramatically reduces branch-and-bound nodes
   • Don't force SOS on unordered sets

3. Exploit Convexity

   • Check if piecewise linear functions are convex/concave
   • If yes, solve as LP (no MIP needed)

4. Minimize Binary Variables

   • Each binary variable roughly doubles potential search nodes
   • Look for reformulations that reduce binary count

5. Preprocessing

   • Tighten variable bounds before modeling
   • Enables smaller Big-M values
   • Improves solver performance

Implementation in Modeling Languages

AIMMS

• Use Property attribute for SOS1/SOS2 constraints
• Solver automatically recognizes and exploits SOS structure

AMPL/Pyomo/JuMP

• Explicitly declare SOS sets
• Solver interfaces pass SOS information to solver

GAMS

• Special syntax for SOS1/SOS2 sets
• Integrated with major solvers

Further Reading

See reference files for detailed examples:

• references/indicator_variables.md - Binary indicator techniques
• references/either_or_conditional.md - Logical constraints
• references/sos_sets.md - Special Ordered Sets
• references/piecewise_linear.md - Nonlinear approximation
• references/product_elimination.md - Linearization techniques
• references/examples.md - Complete worked examples

References

Based on Chapter 7 "Integer Linear Programming Tricks" from AIMMS Modeling Guide (AIMMS 4, 1993-2018)

Key sources:

• E.M.L. Beale and J.A. Tomlin (1969) - Special Ordered Sets
• H.P. Williams (1990) - Model Building in Mathematical Programming