T03: add LP/MIP modeling tutorial slides

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

Author: d-krupke

slides/T03/T03-lp-mip-modeling.qmd

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+---
+title: "LP/MIP Modeling"
+subtitle: "Algorithm Engineering — T03"
+author: Dr. Dominik Krupke
+format:
+  revealjs:
+    slide-number: true
+    width: 1600
+    height: 900
+    chalkboard:
+      buttons: false
+    view-distance: 999
+    preview-links: auto
+    css: styles.css
+    footer: "Algorithm Engineering SS 2026 — T03 LP/MIP Modeling"
+    progress: true
+    transition: none
+    background-transition: none
+    theme: dark
+    highlight-style: github
+    code-overflow: wrap
+bibliography: references.bib
+---
+
+# Welcome {background-color="#2d4059"}
+
+<!-- Placeholder opener for TTS preview — to be replaced by a filmed intro. -->
+
+{{< include _00-intro.qmd >}}
+
+{{< include _01-mathopt.qmd >}}
+
+{{< include _02-lp-preserving.qmd >}}
+
+{{< include _03-bigm.qmd >}}
+
+{{< include _04-logic.qmd >}}
+
+{{< include _05-activation.qmd >}}
+
+{{< include _06-pwla.qmd >}}
+
+{{< include _07-strong-weak.qmd >}}
+
+
+# See you next time {background-image="assets/symbol_clockwork.png" background-opacity="0.3" background-size="cover" background-color="#2d4059"}
+
+<!-- Placeholder closer for TTS preview — to be replaced by a filmed outro. -->
+

slides/T03/_00-intro.qmd

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+<!-- Sources for Intro:
+     - Slides 1-5 of PLAN/08_SLIDE_SKELETON.md
+     - Bead 1 (LP/MIP as problem classes + naming conventions): PLAN/06_BEADS.md
+     - Bead 3 (Guessing game): PLAN/06_BEADS.md
+     - Bead 4 (Pattern recognition / Box framing): PLAN/06_BEADS.md
+     - Active element on slide 4: PLAN/10_ACTIVE_ELEMENTS.md §10.1
+     - MathOpt (Bead 2) lives in _01-mathopt.qmd
+-->
+
+# LP/MIP Modeling {background-image="assets/symbol_lpmip.png" background-opacity="0.3" background-size="cover" background-color="#2d4059"}
+
+<!-- Story Line:
+1. There are these two problem classes, LP and MIP, with powerful solvers. Which problems can we solve with them?
+2. Here are the canonical base problems each class handles, and the industrial patterns they map onto.
+3. Speaking to these solvers is quite easy, and the solvers are interchangeable.
+4. More expressions than you might think are actually linear.
+5. Tension between accuracy and tractability.
+-->
+## Two Problem Classes with Powerful Solvers
+
+[**Linear Expressions**]{style="display:block; text-align:center;"}
+
+<svg viewBox="0 0 400 60" preserveAspectRatio="none" style="width:100%; height:60px; display:block;" aria-hidden="true">
+  <defs>
+    <marker id="arrowhead" markerWidth="8" markerHeight="8" refX="4" refY="4" orient="auto">
+      <path d="M0,0 L8,4 L0,8 z" fill="currentColor"/>
+    </marker>
+  </defs>
+  <path d="M 200 0 L 200 25 L 96 25 L 96 50" stroke="currentColor" stroke-width="2" fill="none" marker-end="url(#arrowhead)"/>
+  <path d="M 200 0 L 200 25 L 304 25 L 304 50" stroke="currentColor" stroke-width="2" fill="none" marker-end="url(#arrowhead)"/>
+</svg>
+
+
+
+:::: {.columns}
+::: {.column width="48%"}
+$$
+\begin{aligned}
+\max\quad        & \sum_{e \in \delta^+(s)} x_e \\
+\text{s.t.}\quad & \sum_{e \in \delta^+(v)} x_e = \sum_{e \in \delta^-(v)} x_e & \forall v \in V \\
+                 & 0 \le x_e \le u_e & \forall e \in E
+\end{aligned}
+$$
+
+:::
+::: {.column width="4%"}
+:::
+::: {.column width="48%"}
+
+$$
+\begin{aligned}
+\max\quad        & \sum_i v_i\, b_i \\
+\text{s.t.}\quad & \sum_i w_i\, b_i \le W \\
+                 & b_i \in \{0, 1\} && \forall i
+\end{aligned}
+$$
+
+:::
+::::
+:::: {.columns}
+::: {.column width="48%"}
+
+[Continuous Variables (Linear Programs)]{style="display:block; text-align:center;"}
+
+[*Solvable in polynomial time.*]{style="color:#6aa84f; display:block; text-align:center;"}
+:::
+::: {.column width="4%"}
+:::
+::: {.column width="48%"}
+
+[Discrete Variables (Mixed-Integer Programs)]{style="display:block; text-align:center;"}
+
+[*NP-hard in general but often tractable.*]{style="color:#c0392b; display:block; text-align:center;"}
+:::
+::::
+
+::: {.fragment}
+::: {.platypus-question}
+For both problem classes, there are highly engineered solvers. How much can we express in each class? How do we know if it is the right technology?
+:::
+:::
+
+## The diet problem
+
+:::: {.columns}
+
+::: {.column width="38%"}
+$$
+\begin{aligned}
+\min\quad        & \sum_j c_j\, x_j \\
+\text{s.t.}\quad & \sum_j a_{ij}\, x_j \;\ge\; b_i & \forall i \\
+                 & x_j \ge 0
+\end{aligned}
+$$
+
+*Choose food quantities $x_j$ to meet every nutrient requirement $b_i$ at minimum cost. Coefficient $a_{ij}$ = nutrient $i$ per unit of food $j$.*
+:::
+
+::: {.column width="4%"}
+:::
+
+::: {.column width="58%"}
+
+::: {.compact-table}
+| Food $j$ | kcal | protein (g) | calcium (mg) | $c_j$ (€/unit) |
+|---|---:|---:|---:|---:|
+| Oats  | 380 | 13 |  50 | 0.20 |
+| Milk  | 150 |  8 | 300 | 0.80 |
+| Eggs  |  80 |  7 |  30 | 0.30 |
+| Beans | 230 | 15 |  80 | 0.40 |
+| **Daily req. $b_i$** | **2000** | **50** | **800** | |
+:::
+
+---
+
+::: {.fragment}
+Real-world siblings:
+
+- **Feed formulation**: livestock rations from grains
+- **Alloy blending**: composition spec from raw metals
+- **Fuel mix**: octane rating from crude streams
+- **Fertilizer mixing**: N/P/K from base products
+- **Production mix**: products from limited resources
+:::
+
+:::
+
+::::
+
+## Network Flow
+
+:::: {.columns}
+
+::: {.column width="48%"}
+$$
+\min\quad        \sum_{e \in E} c_e\, x_e
+$$
+$$
+\sum_{e \in \delta^-(v)} x_e - \sum_{e \in \delta^+(v)} x_e = b_v\quad  \forall v \in V \\
+                  0 \le x_e \le u_e \quad \forall e  \in E
+$$
+
+![](assets/network_flow.png){fig-align="center" width="100%"}
+:::
+
+
+::: {.column width="52%"}
+::: {.fragment}
+Real-world siblings:
+
+- **Shipping & logistics**: origins to destinations
+- **Telecom routing**: link capacities, endpoint demand
+- **Power dispatch**: generation, load, transmission
+- **Supply chains**: multi-stage production
+- **Pipeline scheduling**: capacitated arcs
+
+Special cases: **shortest path**, **max-flow**, **transportation**, **assignment**.
+:::
+
+---
+
+*Edge labels: flow $x_e$ / capacity $u_e$. $b_v < 0$ supplies, $b_v > 0$ demands.*
+:::
+
+::::
+
+## Matchings/Assignments
+
+:::: {.columns}
+
+::: {.column width="38%"}
+$$
+\begin{aligned}
+\max\quad        & \sum_{e \in E} c_e\, x_e \\
+\text{s.t.}\quad & \sum_{e \in \delta(v)} x_e \;\le\; 1 && \forall v \in V \\
+                 & x_e \in \{0,1\}
+\end{aligned}
+$$
+
+*Pick edges of an undirected graph to maximize total weight, no two chosen edges sharing a vertex. A vertex may stay unmatched.*
+:::
+
+::: {.column width="4%"}
+:::
+
+::: {.column width="58%"}
+::: {.fragment}
+Real-world siblings:
+
+- **Roommate pairing**: dorm allocation
+- **Tournament rounds**: chess Swiss-style pairings
+- **Ride-share pairing**: two riders share a car
+- **Kidney exchange**: donor-recipient pairs
+- **Peer programming**: students paired by skill
+:::
+
+:::
+
+::: {.fragment}
+::: {.platypus-info}
+The basic version is polynomial, but as soon as there are interdependencies between edges or we have t-wise instead of pairwise matchings, the Blossom algorithm fails and we need MIP.
+:::
+:::
+
+::::
+
+## Set Cover
+
+:::: {.columns}
+
+::: {.column width="38%"}
+$$
+\begin{aligned}
+\min\quad        & \sum_{S \in \mathcal{S}} c_S\, y_S \\
+\text{s.t.}\quad & \sum_{S \ni i} y_S \;\ge\; 1 && \forall\ i\\
+                 & y_S \in \{0,1\}
+\end{aligned}
+$$
+
+*Pick subsets $S$ so every element $i$ is covered at least once, at minimum cost.*
+:::
+
+::: {.column width="4%"}
+:::
+
+::: {.column width="58%"}
+::: {.fragment}
+Real-world siblings:
+
+- **Crew scheduling**: duty patterns covering all flights
+- **Facility location**: depots covering all customer regions
+- **Survey routing**: routes covering all neighborhoods
+- **Sensor placement**: cameras covering all zones
+- **Vaccine outreach**: clinics covering all populations
+:::
+
+:::
+
+::: {.fragment}
+::: {.platypus-tip}
+Surprisingly many difficult combinatorial optimization problems turn out to be variants of Set Cover ($\geq 1$), Set Packing ($\leq 1$), or Set Partitioning ($=1$) at heart.
+:::
+:::
+
+::::
+
+## But what if my problem has non-linear expressions?
+
+| # | Expression | $\mathrel{\hat{=}}\text{LP}$ | $\mathrel{\hat{=}}\text{MIP}$ | approx. only |
+|---|---|:-:|:-:|:-:|
+| 1 | $\min\ \max_i\, c_i^\top x_i, x\in \mathbb{R}^n$               | [✓]{.fragment} |                |                |
+| 2 | $\max\ \sum_i \lvert x_i \rvert, x\in \mathbb{R}^n$               |  | [✓]{.fragment} |                |                |
+| 3 | $\min\ \sum_i \lvert x_i \rvert, x\in \mathbb{R}^n$            | [✓]{.fragment} |                |                |
+| 4 | $\min\ \sum_i x_i^2, x\in \mathbb{R}^n$                        |                |                | [✓]{.fragment} |
+| 5 | $c^\top x \ge a\ \ \lor\ \ c'^\top x \le a', x\in \mathbb{R}^n$|                | [✓]{.fragment} |                |
+| 6 | $z = x\,y,\ x \in \{0,1\},\ y \in [L, U]$   |                | [✓]{.fragment} |                |
+| 7 | $z = x\,y,\ x, y \in [L, U]$                |                |                | [✓]{.fragment} |
+
+::: {.platypus-question}
+Things are not always as they seem. Which of these non-linear expressions are actually linear in disguise?
+:::
+
+## "All models are wrong, but some are useful." (George Box)
+
+::: {.incremental}
+- Modeling is a constant tension between **accuracy** and **tractability**.
+- LPs are one of the most benign (optimization) problem classes available.
+     - Highly optimized solvers solving millions of variables in seconds.
+- Well-formulated MIPs are also often quite tractable.
+     - In particular, if the LP relaxation is tight.
+- Detecting tractable modeling patterns is the **critical skill** for building useful models.
+:::
+
+::: {.platypus-confucypus}
+“Every hard problem has a weakness.
+The modeler’s art is to find it before the solver finds despair.”
+
+— Confucypus
+:::
+