Non-linear Optimisation and Portfolio Selection

8.1 Non-linear Programming: Why It Is Harder Than Linear Programming

Linear programming relies on a key proportionality assumption: the contribution of each activity is proportional to its level, so the objective always takes the form “coefficient × decision variable.” When this assumption fails, the problem becomes a non-linear programme.

Linear versus Non-linear

Four Types of Non-proportional Relationships

8.2 Curve Fitting and Multi-Start Solving

How to Obtain a Non-linear Relationship

1. Plot the data as a scatter chart.
2. Add a trendline (Chart Tools → Layout → Trendline).
3. Choose a functional form (linear, logarithmic, polynomial, power or exponential), inspect R² and residuals, and check whether the parameters have economic meaning.

Multi-Start Search: The Same Model, Different “Optima”

Consider Maximise Profit = 0.5x⁵ − 6x⁴ + 24.5x³ − 39x² + 20x subject to 0 ≤ x ≤ 5. It has two local maxima, so a solver starting from different values can stop in different places.

8.3 Two Non-linear Cases

Case 1: Wyndor with Marketing Costs

Market research shows that, without advertising, Wyndor can sell only a small fraction of output; selling everything requires substantial advertising. Curve fitting estimates weekly marketing cost for production rates D and W as: doors 25D²; windows (200/3)W²; gross margin is about USD 375 per door and USD 700 per window. Therefore:

Case 2: Portfolio Selection

Large fund managers commonly use non-linear-programming models to set portfolio weights. Investors care about expected return and risk—a cost-benefit trade-off (non-linear) model:minimise risk subject to expected return meeting a required level.

8.4 Separable Programming and Evolutionary Algorithms

Case: Wyndor with Overtime

Wyndor receives an order for a special handmade product that will occupy Plant 1 and Plant 2 capacity for the next four months. Remaining workers must work overtime to produce standard products. Overtime adds cost, so profit is lower at the same output—a piecewise-linear non-proportional relationship.

Why Use Separable Programming?

• Transform the problem into Linear programming, enabling a fast solution.
• A linear programme can generate a sensitivity report(shadow prices and allowable ranges)—non-linear solvers usually cannot.
• The method segments each activity that violates proportionality, creates one variable per segment, and uses segment slope as unit profit.

Evolutionary Solver and Genetic Algorithms

Evolutionary Solver follows a completely different logic. A standard solver starts from a single solution and searches in improving directions. Evolutionary Solver generates an entire population of solutions and iterates through “pairing → offspring → mutation → survival of the fittest” until several generations show no improvement. It suits rough, irregular models with many local optima.

Evolutionary Algorithms: Logic and Characteristics

The method borrows a biological metaphor: each candidate solution is an individual’s “chromosome,” with its decision values encoded as “genes”;a group of candidate solutions is the population, and the objective value measures each individual’s “fitness.” Each generation does three things:

• Selection: fitter individuals are more likely to reproduce—survival of the fittest.
• Crossover: combine genes from two parents to create offspring. For portfolios, this means exchanging and blending two sets of weights.
• Mutation: randomly change a few genes (for example, increase or decrease a weight slightly). This is crucial for shaking a population out of a local optimum.

Its characteristics determine when it is suitable:

• No gradient required: it only needs to calculate the objective for a set of values, so it handles rough, discontinuous, conditional or integer models—the cases ordinary solvers find hardest.
• Starts from a population rather than one point, reducing the chance of immediate entrapment in a local optimum, but it does not guarantee a global optimum—only a sufficiently good solution.
• Results are random: two runs on the same model may differ. Fix a random seed and compare multiple runs rather than relying on one.
• Parameters must be set: population size, crossover rate, mutation rate, generations and convergence criteria all affect quality and runtime.
• Computationally intensive: the objective is evaluated thousands of times. It is best for problems with relatively few variables but difficult model structure.
• Strong on combinatorial problems: selecting a subset from many candidates (stocks, sites), scheduling with logical constraints and multi-modal curve fitting.

Case: Selecting a Portfolio That Beats the Market

A common objective for investment managers is to beat the market. If past performance has some predictive value, it makes sense to search for portfolios that beat the market most often. In this case, seven A-share stocks (ICBC, PetroChina, Kweichow Moutai, BYD, COSCO Shipping Holdings, China Vanke and Ping An) have returns for the past 12 quarters, along with the SSE A-Share Index over the same period; assign a weight to each stock.

The objective is unusual: it counts wins, not the size of wins. A small change in weights may leave the win count unchanged or jump it by one—the objective is a step function, non-differentiable and discontinuous, with many tied solutions. Ordinary solvers struggle; evolutionary algorithms are appropriate:the population is a set of weight combinations,fitness is the number of winning quarters (plus a small penalty enforcing weights summing to one),Crossover blends two weight vectors, and Mutation randomly changes a few weights slightly.

A simple comparison with this data:the equal-weight portfolio(1/7 in each stock) beat the index in 8 of the past 12 quarters. Blindly generating 200,000 random weight vectors, most beat the index only 6–8 times; only a few dozen reached 11 wins, and only an extremely small number reached 12.This is why we use search rather than luck: evolutionary algorithms accumulate information from each trial and move toward higher fitness over generations.

8.5 AI and the Course Project

When AI Portfolio Recommendations Fail

Course Project Requirements (1): Objective and Deliverables

• Objective: choose a real management problem and complete the full “data — model — AI collaboration — recommendation” process.
• Deliverable: a written report; no classroom presentation.
• Report elements: problem definition, data description, model and method, interpretation and recommendation.
• Required AI-use statement: which tool, for what purpose and how it was verified.
• Topic guidance: extend a course case or address a real problem from your workplace.

Course Project Requirements (2): Report Outline

1. Problem and context: what decision is needed and how will it be measured?
2. Data and definitions: sources, field meanings and cleaning.
3. Model and solution: why this model, and how it is solved.
4. Results and interpretation: what the result is and what it means for management.
5. AI collaboration and verification: what AI did and what problems you found.
6. Recommendation and limitations: what to do, plus boundaries and risks.

8.6 Summary and Course Review

• Non-linearity arises from non-proportional relationships; such models are much harder to solve and offer no guarantee of finding a global optimum.
• Curve fitting and multi-start search are standard tools: different starting values can yield different “optima.”
• Separable programming converts piecewise-linear problems into linear programmes and provides sensitivity reports.
• Evolutionary or genetic algorithms suit rough, irregular models.

The Course Spine across Eight Units

Appendix: Interactive Demonstrations and Classroom Exercises

Data sources: 单元8_非线性与组合练习.xlsx and model parameters from WyndorExtension.xlsx and Portfolio.xlsx. Every optimum is calculated on the page by grid search or enumeration and can be checked against Excel Solver.

Unit Glossary (English–Chinese)