Limits of Mathematical Programming for Systems Analysis

Lecture 22

November 6, 2023

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Review and Questions

Mathematical Programming

  • Write decision problem in closed-form:
    • Objectives, constraints as explicit functions of decision variables.
  • Linear Programming:
    • Linearity
    • Certainty
    • Divisibility

Relaxing LP Assumptions

  • Relaxing Divisibility: Mixed Integer Linear Programming
  • Relaxing Certainty: Two-Stage Linear Programs
  • Relaxing Linearity: not going there

Treatment of Uncertainty in LPs

  • Two-Stage Problems (Initial Decision -> Recourse)
  • Scenarios

Thinking More About Mathematical Programming

Pros and Cons of Mathematical Programming

Pros:

  • Can guarantee analytic solutions (or close)
  • Often computationally scalable (especially for LPs)
  • Transparent

Cons:

  • Often quite stylized
  • Limited ability to treat uncertainty

Challenge 1: Writing Down A Mathematical Program

Systems models often have:

  1. Hard to write objectives and constraints in closed form;
  2. Nonlinear dynamics with “unpleasant” geometry.

Recall: Bifurcations

Bifurcations are when the qualitative behavior of a system changes across a parameter boundary.

Bifurcations Diagram

Recall: Feedback Loops

Feedback loops can be reinforcing or dampening.

Dampening feedback loops are associated with stable equilibria, while reinforcing feedback loops are associated with instability.

Reinforcing Feedback Example

What Are The Implications of These Dynamics?

  • May violate strong geometric constraints from mathematical programming.
  • Neglecting can lead to overconfidence about outcomes.

Emergence

  • Simple “micro”-scale rules can yield more complex and unexpected “macro”-scale outcomes
  • Can complicate writing down system dynamics and constraints in MP form.

Schelling Model Animation

Source: Blaqdolphin - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=91228415

Challenge 2: Multiple Objectives

We often have multiple objectives that we want to analyze.

  • Costs
  • Environmental impacts
  • Reliability

Mathematical Programming and Multiple Objectives

We could formulate constraints based on a priori assessments of acceptability:

  • Budget constraints;
  • Reliability constraints

Combining Multiple Objectives Together

Another common approach is to combine multiple objectives \(Z_i\) together into a weighted sum:

\[\sum_i w_i Z_i,\]

where \(\sum_i w_i = 1\).

This requires placing the \(Z_i\) on a common scale (normalization).

Limits of These Approaches

Can you think of problems or limits with these approaches to handling multiple constraints?

  • Require a priori assessments of acceptable limits and/or weights.
  • Limits ability to understand “macro”-scale tradeoffs (beyond shadow prices).

Example: Shallow Lake Problem

Shallow Lake Problem

Recall the shallow lake problem from earlier in the semester.

Shallow Lake Problem Diagram

Lake Model

\[\begin{align*} X_{t+1} &= X_t + a_t + y_t \\ &\quad + \frac{X_t^q}{1 + X_t^q} - bX_t,\\[0.5em] y_t &\sim \text{LogNormal}(\mu, \sigma^2) \end{align*}\]

Parameter Definition
\(X_t\) P concentration in lake
\(a_t\) point source P input
\(y_t\) non-point source P input
\(q\) P recycling rate
\(b\) rate at which P is lost

Lake Model Dynamics

GKS: could not find font bold.ttf
No Exogenous P Inputs
Exogenous P Inputs: 0.10/time

The Lake Problem As a Mathematical Program?

What complicates formulating a mathematical program for the lake problem?

The Lake Problem As a Mathematical Program?

Our objective might be to maximize \(\sum_{t=1}^T a_t\) (as a proxy for economic activity), while keeping a low probability of eutrophication.

Uncertain Inputs and MP

How can we represent uncertainty in inputs \(y_t\) in an MP?

  • Very challenging to formulate scenarios;
  • Non-trivial (if possible) to write expected value in analytic form;
  • But Monte Carlo simulation is straightforward…

Multiple Objectives and the Lake Problem

What are some relevant objectives for the lake problem?

  • Maximize phosphorous output (proxy for economic output)
  • Maximize probability of avoiding eutrophification
  • Can add other objectives breaking out lake response (stability, worst case P concentration, etc.)

Upshot Of These Challenges

Many systems problems with

  • complex, non-linear dynamics;
  • continuous uncertainties;
  • multiple objectives.

don’t lend themselves well to mathematical programming.

So how can we make decisions?

Key Takeaways

Takeaways

  • General challenges to mathematical programming for general systems analysis.
  • These challenges include:
    • Complex and non-linear systems dynamics;
    • Uncertainties;
    • Multiple objectives.

Upcoming Schedule

Wednesday: Simulation-optimization as an alternative to mathematical programming.

Friday: Lab on simulation-optimization.

Assessments

HW5 and Project Update: Due Friday.