steepest ascent

**Steepest ascent** (or **steepest descent** for minimization) is an optimization technique that uses the results of a **first-order (linear) DOE** to determine the **direction of fastest improvement** in the response, then takes steps in that direction to move rapidly toward the optimal region. **How Steepest Ascent Works** - **Step 1 — Fit a Linear Model**: Run a first-order DOE (e.g., $2^k$ factorial) around the current operating point. Fit a linear model: $\hat{y} = b_0 + b_1 x_1 + b_2 x_2 + ... + b_k x_k$. - **Step 2 — Determine the Gradient**: The coefficients $b_1, b_2, ..., b_k$ define the gradient direction — the direction in factor space where the response increases fastest. - **Step 3 — Take Steps**: Move the operating point in the gradient direction by increments proportional to the effect sizes. The factor with the largest effect takes a full step; others take proportionally smaller steps. - **Step 4 — Evaluate**: Run experiments at each new point along the path. The response should improve at each step. - **Step 5 — Stop**: When the response stops improving (or starts getting worse), you have reached the neighborhood of the optimum — switch to a **response surface design** (RSM) for fine optimization. **Example: Etch Rate Optimization** - Current conditions: Power = 300W, Pressure = 35 mTorr. - DOE results: $\hat{y} = 150 + 40x_1 + 10x_2$ (power has 4× the effect of pressure). - Steepest ascent direction: increase power in large steps, increase pressure in smaller steps. - Path: (300W, 35mT) → (340W, 36mT) → (380W, 37mT) → (420W, 38mT) → ... - Continue until etch rate peaks and starts declining. **Why Not Jump Directly to the Optimum?** - The linear model is only valid near the current operating point — it doesn't predict where the true optimum is. - By taking **incremental steps** and checking the response, you follow the actual response surface rather than trusting a model extrapolated far from its data. - The path adapts to the true response shape, even if it's nonlinear. **Steepest Ascent in the RSM Framework** The full RSM optimization workflow: 1. **Screening DOE** → identify important factors. 2. **First-Order DOE** → fit linear model. 3. **Steepest Ascent** → move toward optimum region. 4. **Second-Order DOE (CCD/BBD)** → fit quadratic model near the optimum. 5. **Optimize** → find exact optimal settings from the quadratic model. **Practical Tips** - **Step Size**: The first step along the path is typically 1–2× the factor range used in the DOE. If the DOE used ±50W for power, the first step might be 50–100W. - **Stop Criterion**: Run 5–10 points along the path. Stop when 2–3 consecutive points show no improvement. - **Curvature Check**: Include center points in the initial DOE — if curvature is already detected, skip steepest ascent and go directly to RSM. Steepest ascent is the **efficient bridge** between screening/factorial designs and response surface optimization — it moves the experimenter quickly from a suboptimal region to the neighborhood of the optimum.