You are an expert in identifying **nonlinear patterns** in mixed-integer optimization problems.

## TASK

Given a LaTeX optimization problem, detect and group similar indexed instances of these nonlinearities:

1. **Bilinear:** Products of decision variables (e.g. $x_i \cdot y_j$)
2. **Min:** Simple min operators (e.g. $\min(x_i, y_i)$)
3. **Max:** Simple max operators (e.g. $\max(x_i, y_i)$)
3. **Absolute Value:** Terms like $|x_i - y_i|$ with linear decision variables
4. **Quotient:** Linear fractional terms where DECISION VARIABLES appear in either numerator or denominator or both (e.g., $\frac{{y_i}}{{x_i}}$ where both $x_i$ and $y_i$ are decision variables).
5. **Monotone Transformation:** Objective functions of form $\min f(g(x))$ where $g(x)$ is linear and $f$ is monotone function (e.g., \log(\sum_i x_i)$)

## NOTE
If a category has no patterns, write "NONE" under that category.
Focus on PATTERNS that represent multiple similar terms, not individual instances.

Report grouped patterns succinctly:

* Example grouping: $x_i \cdot y_j, \forall i,j \in A$
* If none found, explicitly state "NONE"

## INPUT

* **LaTeX model:** {latex_model}
* **Parameter context:** {parameter_context}
* **Concrete parameters (IGNORE these when detecting nonlinearities):** {param_info}

## OUTPUT FORMAT

✅ NON-LINEARITIES DETECTED: [YES/NO]

BILINEAR_PATTERNS:

* [List each bilinear pattern]

MIN_PATTERNS:

* [List each min pattern]

MAX_PATTERNS:

* [List each max pattern]

ABSOLUTE_PATTERNS:

* [List each absolute value pattern]

QUOTIENT_PATTERNS:

* [List each quotient pattern]

MONOTONE_TRANSFORMATION_PATTERNS:

* [List each monotone transformation pattern in objective function]

**IMPORTANT:**

* Provide ONLY mathematical formulations.
* Do NOT use markdown code fences or additional explanation s.