cuopt-numerical-optimization-formulation

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# Numerical Optimization Formulation

Concepts and workflow for going from a problem description to a clear formulation across LP, MILP, and QP. No API code here.

## What is LP / MILP / QP

- **LP**: Linear objective, linear constraints, continuous variables.
- **MILP**: Same as LP plus some integer or binary variables (e.g., scheduling, facility location, selection).
- **QP**: Quadratic objective (e.g., x², x·y terms — portfolio variance, least squares), linear constraints. **QP support in cuOpt is currently in beta.**

## Identifying problem type

| Property | LP | MILP | QP |
|---|---|---|---|
| Objective | Linear | Linear | Quadratic (xᵀQx + cᵀx) |
| Constraints | Linear | Linear | Linear + convex quadratic (inequality only) via second-order cones |
| Variables | Continuous | Mixed: continuous + integer/binary | Continuous |
| Sense | min or max | min or max | **minimize only** (negate to max) |
| Duals / sensitivity | Dual values + reduced costs | **None** (integer optima) | Dual values + reduced costs |

If the objective is purely linear, prefer LP/MILP — do not artificially introduce quadratic terms. If any variable is integer or binary, the problem is MILP regardless of the rest.

**Post-solve sensitivity (LP / QP only).** Continuous LP and QP solutions expose **dual values** (the marginal objective change per unit a binding constraint is relaxed: *where to invest to improve the outcome*) and **reduced costs** (for a variable the optimizer left at zero, how far it must improve to enter the solution: a *near-miss*). **MILP solutions have no duals** — integer optima are not continuous, so there are none to return. Duals are also unavailable when the model includes quadratic constraints — the second-order cone path returns primal values only. See the language-specific API skills for how to retrieve them after a solve.

## Required formulation questions

Ask these if not already clear:

1. **Decision variables** — What are they? Bounds?
2. **Objective** — Minimize or maximize? Linear or quadratic? For QP: any squared or cross terms (x², x·y)? If maximize a quadratic, the user must negate and minimize.
3. **Constraints** — Linear inequalities/equalities? Convex quadratic constraints (inequality only) are also supported, handled as second-order cones; non-convex or equality quadratic constraints are not.
4. **Variable types** — All continuous (LP / QP) or some integer/binary (MILP)?
5. **Convexity (QP only)** — For minimization, the quadratic form (matrix Q) should be positive semi-definite for well-posed problems.

## Typical modeling elements

- **Continuous variables** — production amounts, flow, allocations, portfolio weights.
- **Binary variables** — open/close, yes/no (e.g., facility open, item selected).
- **Linking constraints** — e.g., production only if facility open (Big-M or indicator).
- **Resource constraints** — linear cap on usage (materials, time, capacity).
- **Quadratic objective terms** — variance (xᵀQx), squared error (‖Ax − b‖²), interaction terms.

## Typical QP use cases

- Portfolio optimization — minimize variance subject to return and budget.
- Least squares — minimize ‖Ax − b‖² subject to linear constraints.
- Other quadratic objectives with linear constraints.

---

## Problem statement parsing

When the user gives **problem text**, classify every sentence and then summarize before formulating. The parsing framework below applies regardless of LP / MILP / QP.

**Classify every sentence** as **parameter/given**, **constraint**, **decision**, or **objective**. Watch for **implicit constraints** (e.g., committed vs optional phrasing) and **implicit objectives** (e.g., "determine the plan" + costs → minimize total cost).

**Ambiguity:** If anything is still ambiguous, ask the user or solve all plausible interpretations and report all outcomes; do not assume a single interpretation.

### 🔒 MANDATORY: When in Doubt — Ask

- If there is **any doubt** about whether a constraint or value should be included, **ask the user** and state the possible interpretations.

### 🔒 MANDATORY: Complete-Path Runs — Try All Variants

- When the user asks to **run the complete path** (e.g., end-to-end, full pipeline), run all plausible variants and **report all outcomes** so the user can choose; do not assume a single interpretation.

### Three labels

| Label | Meaning | Examples (sentence type) |
|-------|--------|---------------------------|
| **Parameter / given** | Fixed data, inputs, facts. Not chosen by the model. | "Demand is 100 units." "There are 3 factories." "Costs are $5 per unit." |
| **Constraint** | Something that must hold. May be explicit or **implicit** from phrasing. | "Capacity is 200." "All demand must be met." "At least 2 shifts must be staffed." |
| **Decision** | Something we choose or optimize. | "How much to produce." "Which facilities to open." "How many workers to hire." |
| **Objective** | What to minimize or maximize. May be **explicit** ("minimize cost") or **implicit** ("determine the plan" with costs given). | "Minimize total cost." "Determine the production plan" (with costs) → minimize total cost. |

### Implicit constraints: committed vs optional phrasing

**Committed/fixed phrasing** → treat as **parameter** or **implicit constraint** (everything mentioned is given or must happen). Not a decision.

| Phrasing | Interpretation | Why |
|----------|-----------------|-----|
| "Plans to produce X products" | **Constraint**: all X must be produced. | Commitment; production level is fixed. |
| "Operates 3 factories" | **Parameter**: all 3 are open. Not a location-selection problem. | Current state is fixed. |
| "Employs N workers" | **Parameter**: all N are employed. Not a hiring decision. | Workforce size is given. |
| "Has a capacity of C" | **Parameter** (C) + **constraint**: usage ≤ C. | Capacity is fixed. |
| "Must meet all demand" | **Constraint**: demand satisfaction. | Explicit requirement. |

**Optional/decision phrasing** → treat as **decision**.

| Phrasing | Interpretation | Why |
|----------|-----------------|-----|
| "May produce up to …" | **Decision**: how much to produce. | Optional level. |
| "Can choose to open" (factories, sites) | **Decision**: which to open. | Selection is decided. |
| "Considers hiring" | **Decision**: how many to hire. | Hiring is under consideration. |
| "Decides how much to order" | **Decision**: order quantities. | Explicit decision. |
| "Wants to minimize/maximize …" | **Objective** (drives decisions). | Goal; decisions are the levers. |

### Implicit objectives — do not miss

**If the problem asks to "determine the plan" (or similar) but does not state "minimize" or "maximize" explicitly, the objective is often implicit.** You **MUST** identify it and state it before formulating; do not build a model with no objective.

| Phrasing / context | Likely implicit objective | Why |
|-------------------|---------------------------|-----|
| "Determine the production plan" + costs given (per unit, per hour, etc.) | **Minimize total cost** (production + inspection/sales + overtime, etc.) | Plan is chosen; costs are specified → natural goal is to minimize total cost. |
| "Determine the plan" + costs and revenues given | **Maximize profit** (revenue − cost) | Both sides of the ledger → optimize profit. |
| "Try to determine the monthly production plan" + workshop hour costs, inspection/sales costs | **Minimize total cost** | All cost components are given; no revenue to maximize → minimize total cost. |

**Rule:** When the problem gives cost (or cost and revenue) data and asks to "determine", "find", or "establish" the plan, **always state the objective explicitly** (e.g., "I'm treating the objective as minimize total cost, since only costs are given."). If both cost and revenue are present, state whether you use "minimize cost" or "maximize profit". Ask the user if unclear.

### Parsing workflow

1. **Split** the problem text into sentences or logical clauses.
2. **Label** each: parameter/given | constraint | decision | **objective** (if stated).
3. **Identify the objective (explicit or implicit):** If the problem says "minimize/maximize X", that's the objective. If it only says "determine the plan" (or "find", "establish") but gives costs (and possibly revenues), the objective is **implicit** — state it (e.g., minimize total cost, or maximize profit) and confirm with the user if ambiguous.
4. **Flag implicit constraints**: For each sentence, ask — "Does this state a fixed fact or a requirement (→ parameter/constraint), or something we choose (→ decision)?"
5. **Resolve ambiguity** by checking verbs and modals:

From NVIDIA/skills, Apache-2.0.

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