Multi-objective optimisation compares feasible decisions using several objective functions. A decision is Pareto optimal when no feasible alternative improves at least one objective without worsening another. The Pareto front is the set of objective-value combinations produced by those nondominated decisions.
For a quant comparing risk, expected reward and trading cost, the first payoff is elimination: remove choices that are inferior on the declared criteria. The remaining trade-offs still need a decision maker's preferences. Being on the front does not identify a uniquely best portfolio.
Four choices, three survivors
Suppose a committee can choose exactly one of four hypothetical portfolios. It wants a lower risk score and a higher reward score. Both are stipulated dimensionless scores, not volatility estimates, percentage returns or forecasts. The feasible set consists of these four choices only; combining them is not allowed.
| Choice | Risk score, lower is better | Reward score, higher is better | Verdict |
|---|---|---|---|
| A | 1 | 2 | Nondominated |
| B | 2 | 4 | Nondominated |
| C | 2 | 3 | Dominated by B |
| D | 3 | 5 | Nondominated |
B dominates C: risk is equal at 2, while reward rises from 3 to 4. Requiring a strict improvement in every criterion would mistakenly retain C. Equality on one criterion is compatible with dominance, provided another improves.
A survives because every higher-reward choice also has higher risk. D survives because no other choice reaches its reward of 5. B survives between them: A reduces risk but sacrifices reward, while D raises reward but adds risk. The Pareto front is therefore the three score pairs for A, B and D.
Original finite-menu example. Only the marked choices are feasible; no connecting curve or portfolio mixtures are assumed. All scores are dimensionless illustrations.
The Pareto set contains decisions; the Pareto front contains their objective values. Different holdings can generate the same risk–reward pair, so a single point on a plotted front need not identify a unique allocation. MIT's multiobjective optimisation lecture distinguishes decision space from objective space and separates constructing trade-offs from selecting among them.
For a general minimisation problem with objectives , a feasible decision dominates exactly when
To use that convention here, minimise risk and negative reward. The signs matter: treating both raw scores as costs reverses the committee's reward preference.
A weighted sum makes the preference explicit
One way to select from the menu is scalarisation: replace the vector of objectives with a scalar score. For these illustrative units, maximise
With , the scores for A, B, C and D are respectively 0.5, 1, 0 and 0.5, so B wins. This choice now has a reason beyond nondominance: one extra risk-score unit costs 1.5 reward-score units in the committee's ranking. The coefficient is a preference, not a probability or an inferred market parameter.
For all-minimisation notation, a global minimiser of with every is Pareto optimal. If another decision dominated it, that decision would have a strictly smaller weighted objective. This argument does not require convex optimisation. It does require a global solution; a solver stopping at a local minimum does not establish global nondominance. A zero weight can also let a dominated solution tie when the improvement occurs only in the ignored objective. Boyd and Vandenberghe develop the positive-weight result in their scalarisation slides.
Units change what the same weight means. Multiply every risk score by 100 and leave unchanged: A now wins. To preserve the original ranking, divide the penalty by 100, giving . The Pareto front itself is unchanged by this positive rescaling. Normalising objectives can make weights interpretable, but the chosen scales still help determine the trade-off. John Jasa's OpenMDAO teaching notes explain this scaling issue.
The portfolio-specific efficient frontier is one application: minimise return variance and maximise expected return under stated investment constraints. The Mean–Variance Optimisation lesson develops the corresponding expected-return-minus-variance-penalty formulation. Its public preview outlines the material; the full lesson requires membership.
A Pareto point that every weighted sum misses
Now use a separate fictional menu containing only E, F and G:
| Choice | Risk score | Reward score |
|---|---|---|
| E | 1 | 2 |
| F | 2 | 3 |
| G | 3 | 5 |
All three are nondominated. Yet F can never maximise reward minus a positive risk penalty. To beat or tie E it needs
To beat or tie G it needs
No weight satisfies both conditions. F is an unsupported Pareto point: it is efficient in the feasible menu but cannot be selected by a linear weighted sum.
Original unsupported-point example. The dashed chord is a geometric comparison, not a feasible mixture. Adding more weighted-sum runs cannot recover F.
This is a coverage limitation, not a failure to search a fine enough weight grid. MIT's second multiobjective optimisation lecture explains the problem for nonconvex fronts. If the model instead allowed convex combinations and both scores mixed linearly, the E–G midpoint would have risk 2 and reward 3.5, dominating F. That changes the feasible problem. Actual portfolio risk generally does not mix linearly, so a straight segment between two portfolio score points needs its own justification.
An epsilon-constraint method offers another route: optimise one objective while bounding the others. In this second menu, minimise risk subject to reward being at least 3. E is infeasible; F beats G on risk, so F is selected. Sweeping feasible reward thresholds can reveal trade-offs that weighted sums miss. If risk-minimisers tie, maximise reward among the tied choices before calling the result efficient; otherwise a dominated choice may survive the scalar subproblem. The threshold is a floor, not an instruction to force reward to equal exactly 3. OpenMDAO's epsilon-constraint discussion introduces the objective-to-constraint approach.
The front is conditional on the comparison
Dominance depends on the criteria and feasible alternatives actually included. A choice dominated on risk and reward might survive once trading cost becomes a third objective, if its rival is more expensive. Conversely, an omitted feasible decision may dominate a point in a plotted candidate set. A front computed from a sample is a sample's nondominated subset, unless the search provides evidence that it covers the full problem.
The same distinction matters when inputs are estimated. A front under one set of expected returns, risk forecasts or cost assumptions describes those inputs. It does not prove that the ranking persists under estimation error or that the selected portfolio will deliver a superior realised outcome. Pareto analysis makes the unresolved trade-offs visible; preferences, constraints and input uncertainty still determine how the menu is used.
Sources
- Stephen Boyd and Lieven Vandenberghe, with revised slides by Parth Nobel, Convex Optimization, Chapter 4 slides, especially slides 4.57–4.66: Pareto optimality, portfolio objectives and positive-weight scalarisation.
- Anas Alfaris, Multiobjective Optimization I, MIT MSDO Lecture 14, Spring 2010, slides 18–23: decision versus objective space, preference timing and objective scaling.
- Anas Alfaris, Multiobjective Optimization II, MIT MSDO Lecture 15, Spring 2010, slide 7: unsupported nonconvex trade-offs and the limits of weight refinement.
- John Jasa, NASA Glenn, “Multiobjective optimization,” Practical MDO, sections “Weighted sum method,” “Pareto front” and “Epsilon constraint method”: scaling and repeated scalar optimisation. The finite-menu examples and both figures here are original.