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.