Lecture

Mon., Feb. 19

Today

Overview

Optimization in the wild

Wrapup

- Objective function
- Constraints
- Decision variables

**Reflect**

To what extent is this [in]consistent with exploratory modeling?

**Important**

Today we’ll say *optimization* but even an exact solution is only optimal in our model, not the real world. I prefer the term *policy search* which emphasizes the use of computers to suggest promising strategies.

Today

Overview

Optimization in the wild

Wrapup

**Reflect**

Take 2-3 minutes, then share.

Find a vector \(\mathbf{x}\) that maximizes \(c^T \mathbf{x}\) subject to \(A \mathbf{x} \leq \mathbf{b}\) and \(\mathbf{x} \geq 0\) (Wikipiedia)

**Limitations:**requires strong assumptions (is linearizing your function a good approximation?)**Strengths:**very fast (can scale to large problems)**Examples:**how much should each pump in a water distribution network be run at a given time step to maintain pressure?

- Fixed costs create discontinuities in the objective function
- Example: which electricity generators should be on/off?
- Need to create new indicator variables which flag on/off status: \(\mathbb{I}_i = \begin{cases} 0 & \textrm{off} \\ 1 & \textrm{on} \end{cases}\).
- Can be solved with mixed-integer linear programming (MILP)

If you have a differentiable function, you can use gradient descent to find the minimum.

\[ \mathbf{x}_{n+1} = \mathbf{x}_n - \alpha \nabla f(\mathbf{x}_n) \]

See illustration here

**Strengths:**can handle complex, non-linear systems (model can be a black box)**Limitations:**slow (“guess and check”), rely on “heuristics” to decide a solution is good enough**Examples:**design of water resource systems under uncertainty

**Tip**

We’ll use simulation-optimization in our lab next week.

Today

Overview

Optimization in the wild

Wrapup

- Optimization can be used at a high level (e.g., system design) or can be embedded in a problem (e.g., operations at each time step).
- Every optimization problem has an objective and decision variables. Many have constraints.
- Optimization is a field, with many techniques.
- In this course, I want you to understand and critique how optimization problems are framed in the wild. Take other courses to focus on the techniques.

We will discuss @Schwetschenau et al. (2023). Discussion questions are posted. Please focus on the framing of the problem and the assumptions made when you read; don’t get bogged down in the technical details.

Schwetschenau, S. E., Kovankaya, Y., Elliott, M. A., Allaire, M., White, K. D., & Lall, U. (2023). Optimizing Scale for Decentralized Wastewater Treatment: A Tool to Address Failing Wastewater Infrastructure in the United States. *ACS ES&T Engineering*, *3*(1), 1–14. https://doi.org/10.1021/acsestengg.2c00188