id stringclasses 5
values | category stringclasses 3
values | prompt stringclasses 5
values | model_output stringclasses 5
values | generation_time_sec float64 79 112 | output_token_count int64 1.45k 1.5k |
|---|---|---|---|---|---|
cqp_01 | Constrained Quadratic Program | Minimize f(x, y) = 3*x^2 + 2*y^2 - 4*x + 6*y + 7
subject to the inequality constraint: x + 2*y >= 4
and equality constraint: 2*x - y = 1.
Provide a step-by-step mathematical derivation using KKT conditions. State: (1) The Lagrangian, (2) Stationary gradients, (3) Primal/Dual feasibility, (4) Complementary slackness, an... | To solve the given problem using the Karush-Kuhn-Tucker (KKT) conditions, we need to follow these steps:
### Step 1: Formulate the Lagrangian
The Lagrangian for this problem is:
\[ \mathcal{L}(x, y, \lambda_1, \lambda_2) = 3x^2 + 2y^2 - 4x + 6y + 7 + \lambda_1 (x + 2y - 4) + \lambda_2 (2x - y - 1) \]
### Step 2: Com... | 86.27 | 1,500 |
cqp_02 | Constrained Quadratic Program | Minimize f(x1, x2) = 0.5*x1^2 + x2^2 - 2*x1 - 3*x2
subject to: x1 + x2 <= 2 and x1 >= 0, x2 >= 0.
Derive the solution using KKT conditions. Show all case testing for active vs inactive inequality constraints. | To solve the given optimization problem using the Karush-Kuhn-Tucker (KKT) conditions, we need to follow these steps:
### Problem Formulation:
We are given the objective function:
\[ f(x_1, x_2) = \frac{1}{2}x_1^2 + x_2^2 - 2x_1 - 3x_2 \]
Subject to the constraints:
\[ g_1(x_1, x_2) = x_1 + x_2 - 2 \leq 0 \]
\[ g_2(x... | 111.82 | 1,500 |
pvm_01 | Portfolio Variance Minimization | Minimize portfolio variance: sigma^2 = 2*w1^2 + 3*w2^2 + w3^2 + 2*w1*w2
subject to:
1) Target return: 0.10*w1 + 0.15*w2 + 0.08*w3 = 0.12
2) Budget constraint: w1 + w2 + w3 = 1
Derive the 5x5 system of linear equations using Lagrange multipliers. Solve the linear system step-by-step and provide exact fractional values f... | To solve this problem using the method of Lagrange multipliers, we first define our objective function \( f(w_1, w_2, w_3) = 2w_1^2 + 3w_2^2 + w_3^2 + 2w_1w_2 \) and the constraints:
1. \( g_1(w_1, w_2, w_3) = 0.10w_1 + 0.15w_2 + 0.08w_3 - 0.12 = 0 \)
2. \( g_2(w_1, w_2, w_3) = w_1 + w_2 + w_3 - 1 = 0 \)
The Lagrangi... | 83.93 | 1,500 |
pvm_02 | Portfolio Variance Minimization | Minimize variance: sigma^2 = w1^2 + 2*w2^2 + 1.5*w3^2
subject to: w1 + w2 + w3 = 1 and 0.05*w1 + 0.10*w2 + 0.12*w3 = 0.09.
Solve via matrix inversion or Gaussian elimination showing every arithmetic step. | To solve the problem of minimizing the variance \(\sigma^2 = w_1^2 + 2w_2^2 + 1.5w_3^2\) subject to the constraints \(w_1 + w_2 + w_3 = 1\) and \(0.05w_1 + 0.10w_2 + 0.12w_3 = 0.09\), we will use the method of Lagrange multipliers.
First, let's define the Lagrangian function:
\[ \mathcal{L}(w_1, w_2, w_3, \lambda_1, \... | 84.95 | 1,500 |
nlb_01 | Non-linear Boundary Optimization | Minimize f(x1, x2) = (x1 - 4)^2 + (x2 - 4)^2
subject to:
g1(x): x1 + x2 <= 4
g2(x): x1 - 3*x2 <= 1
x1 >= 0, x2 >= 0
Examine all candidate boundary combinations for KKT multipliers. Verify whether mu_1 >= 0 and mu_2 >= 0 hold for each candidate point. | To solve the given constrained optimization problem using the Karush-Kuhn-Tucker (KKT) conditions, we need to follow these steps:
1. **Formulate the Lagrangian Function:**
The Lagrangian function \( \mathcal{L}(x_1, x_2, \mu_1, \mu_2) \) is given by:
\[
\mathcal{L}(x_1, x_2, \mu_1, \mu_2) = (x_1 - 4)^2 + (x_... | 79 | 1,453 |
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Check out the documentation for more information.
Technical Challenge: Symbolic Drift & Constraint Decay in Frontier LLMs
1. Executive Summary & Background
When deploying AI for core engineering, economic, and mathematical optimization tasks (such as Karush-Kuhn-Tucker conditions or Lagrangian multipliers), calculations require rigorous multi-step invariant tracking. Drawing from practical experience building optimization pipelines, we identify an underexplored capability gap in current models: Symbolic Drift and Constraint Decay.
Unlike human analysts who maintain external working memory or strict symbolic invariants, autoregressive language models treat derivations as next-token distributions. As derivation length increases:
- Coefficients mutate silently across intermediate paragraphs.
- Constraints vanish or fail to enforce complementary slackness (\mu \ge 0).
- Algebraic shortcuts mask missing intermediate arithmetic steps, leading to high residual constraint violations.
2. Systematic Evaluation Setup
- Model Evaluated:
Qwen/Qwen2.5-3B-Instruct(~3.09 Billion parameters, hosted on Hugging Face). - Environment: Google Colab (Free T4 GPU runtime, FP16 precision via Hugging Face Transformers & PyTorch).
- Evaluation Suite: Custom dataset of multi-step constrained quadratic programs and portfolio risk minimization problems requiring exact rational fraction arithmetic and boundary verification.
3. Empirical Observations & Failure Modes
| Problem Category | Prompt Focus | Observed Model Behavior | Evaluation Finding |
|---|---|---|---|
| Constrained Quadratic Program | KKT conditions, stationary equations, primal equality (2x - y = 1) | Generated clean step-by-step algebraic derivations and correct final coordinates (x^*, y^*) = (1.2, 1.4). | Arithmetic Hallucination / Drift: Detailed case-study analysis below reveals subtle arithmetic failure despite clean formatting. |
| Portfolio Variance Minimization | 5x5 linear system solving via Lagrange multipliers | Skipped explicit matrix row-reduction steps and jumped straight to final weight distribution. | Algebraic Shortcut Fallacy: Final weights violated the target expected return constraint by over 14% due to skipped simultaneous equation steps. |
| Non-linear Boundary Optimization | Active vs. inactive inequality case checking | Dropped secondary inequality constraint g_2(x) \le 1 halfway through the derivation. | Constraint Decay: Evaluated an interior saddle point rather than verifying true global boundary feasibility. |
4. Detailed Case Study: Concrete Failure Trace Analysis (cqp_01)
To understand how the model fails under the hood, we audited the generated text for test case cqp_01:
The Model's Optimal Point: The model successfully derived the coordinate candidate (x^*, y^*) = (6/5, 7/5) = (1.2, 1.4).
Primal Constraint Verification:
- Testing equality (2x - y = 1): 2(1.2) - 1.4 = 2.4 - 1.4 = 1.0 (Satisfied).
- Testing inequality (x + 2y \ge 4): 1.2 + 2(1.4) = 1.2 + 2.8 = 4.0 (Meets boundary equality).
The Smoking Gun (Arithmetic Hallucination & Symbolic Strain): When evaluating the objective function f(x, y) = 3x^2 + 2y^2 - 4x + 6y + 7 at this point using rational fractions, the model generated:
[ = 3(36/25)^2 + 2(49/25)^2 - 24/5 + 42/5 + 7 ][ = 108/25 + 98/25 - 120/25 + 210/25 + 175/25 ][ = 491/25 = 19.64 ]The Audit: Summing the numerators correctly yields 108 + 98 - 120 + 210 + 175 = 471, not 491. The model suffered from a hidden arithmetic hallucination at the final summation stage while maintaining flawless linguistic and markdown formatting. This proves that models can mimic proof structures perfectly while failing multi-step arithmetic invariants.
5. Proposed Path Forward
To bridge this gap without scaling model size, we propose:
- Neuro-Symbolic Sandboxing (RLVR): Fine-tuning models via Reinforcement Learning with Verifiable Rewards to output executable symbolic code blocks (e.g.,
SymPy) for intermediate arithmetic and constraint verification. - State-Tracking Attention Registers: Architectural buffers dedicated to pinning variable definitions and active inequalities across deep context windows to prevent symbolic decay.
6. Repository Artifacts
run_eval.py: Complete Google Colab execution script.evaluation_results.json: Raw model outputs, token counts, and execution logs across all categories.
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