Multi-Objective Optimization

Lecture

James Doss-Gollin

Monday, April 6, 2026

Draft: This content is under development.

The Problem with Weights

In Week 9, HCFCD evaluated flood control projects across 7 criteria using weighted scores.

Where do the weights come from?

• Weights reflect values — whose values?
• Different stakeholders have different priorities
• Aggregation requires weights whether they are stated explicitly or embedded in a formula

Target: 1:05. Spend 5 min on this hook. The goal is to motivate multi-objective framing as a way to make value trade-offs explicit rather than hiding them in aggregation. Ask: “Raise your hand if you’d feel comfortable defending the choice of w₁ = 0.3 to a community that gets flooded.” Then pivot: “What if instead we showed decision-makers the full range of trade-offs and let them choose?”

From Single to Multiple Objectives

Today

1. From Single to Multiple Objectives

2. Dominance and Pareto Optimality

3. Weighted Sum and Its Limits

4. Multi-Objective Evolutionary Algorithms

5. Reading and Using the Pareto Front

6. When to Use Multi-Objective Framing

Target: 1:08. Transition into formal setup.

Single-Objective Optimization

Benefit-cost analysis frames adaptation as a single-objective problem:

\[a^* = \arg\max_a \; \mathbb{E}[\text{NPV}(a, \mathbf{s})]\]

This requires that all values can be monetized and aggregated.

Many adaptation objectives — equity, ecosystem health, cultural preservation — resist monetization.

Target: 1:11. Quick slide. Emphasize that BCA is not wrong — it’s a tool with assumptions. When those assumptions hold (everything can be monetized, one decision-maker, shared values), BCA is great. Multi-objective framing is for when they don’t.

Multiple Conflicting Objectives

A multi-objective problem has several objectives that cannot all be maximized simultaneously:

\[\max_a \; \{f_1(a),\; f_2(a),\; \ldots,\; f_k(a)\}\]

Coastal adaptation example:

• \(f_1(a)\): minimize expected flood damages
• \(f_2(a)\): minimize construction cost
• \(f_3(a)\): maximize equity — benefits to vulnerable communities

Improving on \(f_1\) generally requires accepting worse \(f_2\). There is no single “best” solution.

Target: 1:15. Spend about 4 min here. Draw on board if helpful: two axes, each a different objective. Ask students: “Can you think of another pair of objectives in climate adaptation that are in tension?” (e.g., speed of deployment vs. community buy-in; near-term cost vs. long-term resilience).

Dominance and Pareto Optimality

Today

1. From Single to Multiple Objectives

2. Dominance and Pareto Optimality

3. Weighted Sum and Its Limits

4. Multi-Objective Evolutionary Algorithms

5. Reading and Using the Pareto Front

6. When to Use Multi-Objective Framing

Target: 1:18. This is the conceptual core. Take 10 min here.

What Does “Better” Mean?

With multiple objectives we need a new comparison rule.

Dominance: Solution \(a\) dominates solution \(b\) if:

• \(a\) is at least as good as \(b\) on every objective
• \(a\) is strictly better than \(b\) on at least one objective

If \(a\) dominates \(b\), there is no reason to choose \(b\) — you can do better.

Target: 1:21. Walk through this carefully. Emphasize: “at least as good on ALL, strictly better on AT LEAST ONE.” This is a partial order — many solutions won’t be comparable (neither dominates the other).

Dominance: Example

Three house elevation strategies, two objectives (minimize both):

Strategy	Expected Damages (\(M) | Construction Cost (\)M)
A: Elevate 1 ft	4.0	0.5
B: Elevate 3 ft	2.0	1.5
C: Elevate 5 ft	1.8	3.0

• Does A dominate B? No — A is cheaper but has higher damages.
• Does B dominate C? Yes — B is better on cost (1.5 < 3.0) and nearly as good on damages (2.0 vs 1.8)… wait, 2.0 > 1.8, so B is worse on damages. No — neither dominates.
• The Pareto front is {A, B, C} — all three are non-dominated.

Target: 1:23. Walk through the table row by row. Let students work out the dominance comparisons before you reveal. The “wait” moment is intentional — students often jump to conclusions about dominance. Make them check both objectives carefully.

Dominance: Visualized

let
    points = [(2, 2), (1, 4), (4, 1), (3, 3), (5, 2)]
    labels = ["1", "2", "3", "4", "5"]
    colors = [:steelblue, :steelblue, :steelblue, :tomato, :steelblue]

    fig = Figure(; size=(500, 450))
    ax = Axis(
        fig[1, 1];
        xlabel=L"$f_1$ (maximize)",
        ylabel=L"$f_2$ (minimize)",
        xticksvisible=false,
        xticklabelsvisible=false,
        yticksvisible=false,
        yticklabelsvisible=false,
        limits=(0, 6, 0, 5.5),
    )
    for (i, (x, y)) in enumerate(points)
        scatter!(ax, [x], [y]; markersize=22, color=colors[i])
        text!(ax, x - 0.2, y + 0.25; text=labels[i], fontsize=18, font=:bold)
    end

    # Arrow from 1 to 4 showing dominance
    arrows!(ax, [2.15], [2.15], [0.65], [0.65]; linewidth=1.5, color=:gray50)
    text!(ax, 2.7, 2.85; text="dominates", fontsize=12, color=:gray50, rotation=π / 4)

    fig
end

┌ Warning: `arrows` are deprecated in favor of `arrows2d` and `arrows3d`.
└ @ Makie ~/.julia/packages/Makie/kJl0u/src/basic_recipes/arrows.jl:166

Five candidate solutions. Point 4 (red) is dominated by point 1: worse on both objectives. Points 1, 2, 3, 5 are mutually non-dominated.

Target: 1:25. Cold call 2-3 students before revealing. Key: 1 dominates 4 (better on both — higher \(f_1\), lower \(f_2\)). No other dominance pairs among 1, 2, 3, 5. Point 4 (red) is dominated. Ask: “Which points form the Pareto front?” → {1, 2, 3, 5}.

The Pareto Front

The Pareto front is the set of all non-dominated solutions.

• No solution on the front can be improved on one objective without worsening another
• Reveals trade-offs without requiring weights
• The “efficient frontier” of feasible performance

. . .

The Pareto front identifies which solutions are worth considering. Choosing among them is a separate decision that requires values.

Figure 1: A Pareto front in two objectives. Source: Smith et al. (2022)

Target: 1:29. Analogy: the efficient frontier in portfolio theory (risk vs. return). Emphasize the division of labor: analysts find the front, decision-makers choose from it. This is a more honest process than embedding values in weights.

Weighted Sum and Its Limits

Today

1. From Single to Multiple Objectives

2. Dominance and Pareto Optimality

3. Weighted Sum and Its Limits

4. Multi-Objective Evolutionary Algorithms

5. Reading and Using the Pareto Front

6. When to Use Multi-Objective Framing

Target: 1:31. Short section — 5 min.

Can Weighted Sum Find the Pareto Front?

We can scalarize the problem:

\[\min F(a) = \sum_i w_i f_i(a)\]

Sweeping weights \(w_i\) can recover parts of the Pareto front — but not all of it.

• Weighted sum only finds solutions on the convex hull of the front
• Non-convex regions (common in engineering problems) are unreachable by any weighting
• You can miss important solutions entirely

Target: 1:34. The key takeaway: weighted sum is a useful approximation but has fundamental limits. Multi-objective evolutionary algorithms avoid this problem.

Convex vs. Non-Convex Fronts

Convex: weighted sum works

Non-convex: weighted sum misses the red region

Figure 2: From Sudhoff, “Engineering Analysis and Design Using Genetic Algorithms,” Purdue.

Target: 1:37. Walk through both panels. Left: any w₁/w₂ combination traces the full front. Right: the concave region (red) has no tangent weight vector — weighted sum can never find those solutions. Ask: which one do you think is more common in real engineering problems? (Non-convex, generally.)

Multi-Objective Evolutionary Algorithms

Today

1. From Single to Multiple Objectives

2. Dominance and Pareto Optimality

3. Weighted Sum and Its Limits

4. Multi-Objective Evolutionary Algorithms

5. Reading and Using the Pareto Front

6. When to Use Multi-Objective Framing

Target: 1:38. Students don’t need to implement MOEAs — just understand the conceptual machinery well enough to interpret results.

How MOEAs Work

Multi-objective evolutionary algorithms (MOEAs) search for an approximation of the full Pareto front simultaneously.

Core ideas:

• Maintain a population of candidate solutions that evolves over generations
• Selection favors non-dominated solutions (dominance-based ranking)
• Diversity preservation spreads solutions across the front — not just one cluster
• Repeat until convergence

Result: a set of solutions approximating the full trade-off curve — including non-convex regions.

Target: 1:42. Analogy: each solution is a “design.” The MOEA runs many designs in parallel and iteratively improves them. The diversity mechanism is key: without it, the algorithm collapses to one region of the front.

Common MOEAs in Climate/Water Work

• NSGA-II: Classic; uses crowding distance for diversity
• Borg: Widely used in water resources; adaptive population, epsilon-dominance archiving
• Both widely used; Borg tends to perform better on complex, many-objective problems

Convergence diagnostics:

• Repeat with different random seeds — do results change significantly?
• Hypervolume indicator: measures how much of the objective space the front covers
• Compare to weighted-sum benchmarks to detect missed regions

Target: 1:45. These algorithms are heuristics — they don’t guarantee a global optimum. The diagnostics are how we build confidence. Hypervolume is the standard metric: larger = better coverage of the front.

What Does Many-Objective Look Like?

With 2 objectives: a curve.

With 3+ objectives: a surface — impossible to visualize directly.

Parallel axis plots show all objectives simultaneously.

Each line = one non-dominated solution.

Figure 3: Parallel axis plot of non-dominated solutions across six objectives. Source: Kasprzyk et al. (2013)

Target: 1:47. Walk through the parallel axis plot: each vertical axis is one objective (cost, reliability, market use), each colored line is a solution. Solutions that do well on reliability tend to do worse on cost — that’s the trade-off. Color = which model case. This is from a water portfolio problem in the Rio Grande Valley. Point out: 6 objectives, ~1000 non-dominated solutions.

Reading and Using the Pareto Front

Today

1. From Single to Multiple Objectives

2. Dominance and Pareto Optimality

3. Weighted Sum and Its Limits

4. Multi-Objective Evolutionary Algorithms

5. Reading and Using the Pareto Front

6. When to Use Multi-Objective Framing

Target: 1:48. Quick section — 5 min.

Reading a Pareto Front

On a Pareto front plot, each axis is one objective and each point is a non-dominated solution.

The “knee” of the front:

• Region of high curvature — large gains on one objective for small losses on another
• Often a practical default if no strong preference exists
• Not always present; depends on problem structure

The analyst generates the front; the decision-maker selects a solution from it.

Target: 1:50. Preview Lab 10: students will generate a Pareto front for the house elevation problem (cost vs. expected damages) and identify the knee. This is directly connected to Memo 2 — does your project have multiple objectives that BCA collapses?

How Many Objectives?

More objectives → richer problem representation, but real costs:

• Computational: exponentially more solutions needed to cover the front
• Conceptual: harder to reason about trade-offs in high dimensions
• Communication: harder to show decision-makers and stakeholders

Rule of thumb: use as few objectives as possible. Combining objectives into a scalar is often a good idea — when the assumptions hold.

Target: 1:52. This is the counterweight to the earlier motivation. Multi-objective framing is powerful but not free. The question is always: are the objectives genuinely incommensurable, or can they be reasonably combined? Connect to student projects: most have 2–3 meaningful objectives at most.

When to Use Multi-Objective Framing

Today

1. From Single to Multiple Objectives

2. Dominance and Pareto Optimality

3. Weighted Sum and Its Limits

4. Multi-Objective Evolutionary Algorithms

5. Reading and Using the Pareto Front

6. When to Use Multi-Objective Framing

Target: 1:52. Very short section — 1 min.

Choosing the Right Tool

Use multi-objective framing when:

• Stakeholders genuinely disagree on how to weight objectives
• Objectives are incommensurable — not easily reduced to dollars
• You want to make trade-offs explicit and visible to decision-makers

Stick with single-objective (BCA) when:

• One objective clearly dominates
• Stakeholders agree on how to aggregate
• Speed and simplicity matter more than exploring trade-offs

Target: 1:53. Connect to Memo 2: “Look at your project — does your system have conflicting objectives that BCA collapses? If so, multi-objective framing may give a more honest picture.” Don’t frame this as “multi-objective is always better” — it adds complexity and requires more from decision-makers.

Looking Ahead

This week:

• Wednesday: Madison leads discussion of Yang et al. (2023) — come prepared to discuss objectives, algorithm choice, and how results inform decisions
• Friday: Lab 10 — generate and visualize Pareto fronts for the house elevation problem
• Friday: Memo 2 due (Evidence & Robustness)

Week 13 (Monday): Putting it all together — how robustness, scenario discovery, sequential decisions, and multi-objective optimization connect in real projects.

Target: 1:53. End class here. Remind students that Madison is leading discussion on Wednesday — they should read Yang et al. with attention to: (1) what are the objectives? (2) how is the Pareto front generated and interpreted? (3) how do the authors handle trade-offs with decision-makers?

References

Kasprzyk, J. R., Nataraj, S., Reed, P. M., & Lempert, R. J. (2013). Many objective robust decision making for complex environmental systems undergoing change. Environmental Modelling & Software, 42, 55–71. https://doi.org/10.1016/j.envsoft.2012.12.007

Smith, S., Southerby, M., Setiniyaz, S., Apsimon, R., & Burt, G. (2022). Multiobjective optimization and Pareto front visualization techniques applied to normal conducting rf accelerating structures. Physical Review Accelerators and Beams, 25(6), 062002. https://doi.org/10.1103/PhysRevAccelBeams.25.062002

Yang, S., Ruangpan, L., Torres, A. S., & Vojinovic, Z. (2023). Multi-objective Optimisation Framework for Assessment of Trade-Offs between Benefits and Co-benefits of Nature-based Solutions. Water Resources Management, 37(6–7), 2325–2345. https://doi.org/10.1007/s11269-023-03470-8