Portfolio Optimization Under Real Constraints (MILP explained)

Beyond textbook mean-variance optimization: Real-world capital allocation involves constraints that traditional portfolio theory ignores. Grid connection capacity, construction crew availability, ESG quotas, and regulatory timing all matter. Here's how Mixed Integer Linear Programming (MILP) handles the constraints that actually bind your investment committee.

The Problem with Mean-Variance Optimization

Academic portfolio theory assumes you can invest any fraction of your budget in any asset. Want to allocate 23.7% to wind and 31.2% to solar? No problem in theory.

In practice, investment committees face discrete yes/no decisions with complex constraints:

• Indivisible projects: You can't build 60% of a wind farm
• Budget timing: €50M this year, €75M next year, not €125M whenever
• Grid capacity: Only 200MW available at this substation
• Resource limits: Two construction crews, can't do five projects simultaneously
• ESG mandates: Minimum 40% renewables by 2026
• Risk controls: Maximum 15% portfolio VaR

Standard mean-variance optimization breaks down because it can't handle these integer constraints and resource limitations.

Real Constraints in Energy Portfolio Allocation

Let's examine the actual constraints that bind capital allocation decisions in energy and infrastructure:

MILP Formulation: The Mathematics

Mixed Integer Linear Programming elegantly handles both continuous variables (budget allocation) and binary decisions (build/don't build). Here's the core formulation:

Decision Variables

• xᵢ ∈ {0,1}: Binary decision to build project i
• tᵢ ∈ {2024, 2025, ...}: Integer timing decision
• cᵢⱼ ≥ 0: Continuous capital allocation to project i in year j

Key Constraint Types

Efficient Frontier Under Constraints

The traditional efficient frontier shows risk-return combinations for unconstrained portfolios. With MILP, we generate a constrained efficient frontier that respects real-world limitations.

The resulting frontier shows achievable risk-return combinations, not theoretical ones. This gives investment committees realistic options for portfolio construction.

Shadow Prices: Understanding Binding Constraints

MILP solvers provide shadow prices (dual values) that reveal the marginal value of relaxing each constraint. This intelligence transforms capital allocation from guesswork to strategic decision-making.

Shadow prices guide strategic decisions: Should we invest in grid upgrades? Contract additional construction capacity? Which constraints should we negotiate with regulators?

Case Study: German Renewable Portfolio (500MW Budget)

Implementation Best Practices

Data Requirements

• Project financials: NPV, IRR, CAPEX profile, operating cash flows
• Risk metrics: VaR, correlation matrix, sensitivity analyses
• Resource requirements: Grid capacity needs, construction resource timing
• Constraint parameters: Budget envelopes, ESG targets, regulatory limits

Solver Selection

• Commercial: Gurobi, CPLEX (fastest, best support)
• Open source: COIN-OR CBC, GLPK (free, adequate for smaller problems)
• Cloud: Google OR-Tools, Azure Optimization (scalable, no hardware investment)

Model Validation

• Constraint checking: Verify all constraints are satisfied in optimal solution
• Sensitivity analysis: Test how solutions change with constraint variations
• Benchmarking: Compare MILP results with manual allocation approaches
• Stress testing: Ensure robustness under different scenario assumptions

Beyond Basic MILP: Advanced Extensions

Scenario-Robust Optimization

Optimize portfolio under base case assumptions, but ensure feasibility under downside scenarios. This prevents solutions that work on paper but fail in adverse conditions.

Multi-Objective Optimization

Simultaneously optimize multiple objectives (NPV, IRR, ESG score, strategic fit) using weighted objectives or Pareto frontier generation.

Dynamic Programming Integration

Combine MILP portfolio selection with real options valuation to capture both optimal timing and project selection in a unified framework.

Common Pitfalls and How to Avoid Them

The Bottom Line

MILP-based portfolio optimization transforms capital allocation from art to science. By explicitly modeling real-world constraints, investment committees can:

• Make better decisions: Optimal selection considering all binding constraints
• Understand trade-offs: Shadow prices reveal where to focus negotiations
• Achieve higher returns: Systematic optimization typically outperforms intuitive allocation
• Satisfy stakeholders: Demonstrable ESG compliance and risk management

The math is complex, but the value is clear: portfolios optimized under real constraints consistently outperform traditional mean-variance approaches by 100-300 basis points in IRR uplift.