Geometric Programming

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# Introduction to Geometric Programming

@luk036 👨‍💻 · 2026 📅

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#### 📋 Agenda

**Part 1: Fundamentals** 🧠
- What is Geometric Programming?
- Posynomial Functions

**Part 2: GP Transformations** 🔄
- Transformation to Convex
- Design Examples

]
.pull-right[

**Part 3: Applications** 🔬
- EDA Applications
- Circuit Sizing

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### What is Geometric Programming?

Geometric Programming is a class of nonlinear optimization problems that can be transformed into convex optimization problems. The key feature of GP is that it deals with posynomial functions, which are sums of monomials with positive coefficients.

A standard form GP problem is defined as:

$$
\begin{array}{ll}
    \text{minimize} & \quad f_0(x) \\
    \text{subject to} & \quad f_i(x) \leq 1, \quad i = 1, \ldots, m \\
      & \quad h_j(x) = 1, \quad j = 1, \ldots, p
\end{array}
$$

where $f_0, \ldots, f_m$ are posynomials and $h_1, \ldots, h_p$ are monomials.

.mermaid[
<pre>
graph TD
    subgraph "Monomial"
        M["c x₁^{a₁} x₂^{a₂} ⋯ xₙ^{aⁿ}"]
    end
    subgraph "Posynomial"
        P["Σ cₖ x₁^{a₁ₖ} x₂^{a₂ₖ} ⋯ xₙ^{aₙₖ}"]
    end
    M --> P
    P --> GP[GP Constraint fᵢ(x) ≤ 1]
    style M fill:#4caf50
    style P fill:#2196f3
    style GP fill:#ff9800
</pre>
]

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### Key Properties of Geometric Programming

1. **Convexity**: GP problems can be transformed into convex optimization problems through a change of variables.
2. **Global Optimality**: The convex nature ensures that any local optimum is also a global optimum.
3. **Efficient Solving**: GP problems can be solved efficiently using interior-point methods.
4. **Modeling Flexibility**: Many practical constraints can be expressed as posynomial inequalities.

.mermaid[
<pre>
graph LR
    A[GP Problem] --> B["xᵢ = e^yᵢ"]
    B --> C[Convex Problem]
    C --> D[Solve via Interior-Point]
    D --> E["y* → x* = e^y*"]
    style A fill:#ff9800
    style B fill:#2196f3
    style C fill:#4caf50
    style D fill:#9c27b0
    style E fill:#4caf50
</pre>
]

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### Application in Electronic Design Automation

Geometric Programming has found extensive use in EDA due to its ability to model and optimize various aspects of circuit design. Some key applications include:

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### 1. Transistor Sizing

GP can be used to determine optimal transistor sizes in digital and analog circuits. The objective is typically to minimize power consumption or delay while meeting performance constraints.

📚 Example formulation:
- Minimize: Area = $\sum_{i} W_i L_i$
- Subject to:
  - Delay $\leq$ Target_Delay
  - Power $\leq$ Max_Power

Where $W_i$ and $L_i$ are the width and length of transistor $i$.

.mermaid[
<pre>
graph LR
    A[Specifications] --> B[GP Modeling]
    B --> C["Minimize: Σ WᵢLᵢ"]
    C --> D["Subject to: Delay ≤ T_max"]
    D --> E["Subject to: Power ≤ P_max"]
    E --> F[Optimal Transistor Sizes]
    style A fill:#ff9800
    style B fill:#ff9800
    style C fill:#4caf50
    style D fill:#2196f3
    style E fill:#9c27b0
    style F fill:#4caf50
</pre>
]

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### 2. Wire Sizing and Spacing

GP helps in optimizing wire widths and spacing to minimize signal delay and crosstalk while adhering to manufacturing constraints.

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### 3. Power Distribution Network Design

Optimizing the power grid to minimize IR drop and ensure reliable power delivery to all parts of the chip.

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### 5. Analog Circuit Design

GP is particularly useful in analog circuit design for optimizing various performance metrics such as gain, bandwidth, and noise figure.

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### Advantages of GP in EDA

1. **Speed**: GP solvers are typically much faster than general nonlinear optimization methods.
2. **Global Optimality**: Ensures finding the best possible solution within the model's constraints.
3. **Handling Trade-offs**: Easily incorporates multiple, often conflicting, design objectives.
4. **Scalability**: Can handle large-scale problems with thousands of variables and constraints.

.mermaid[
<pre>
graph TD
    A[EDA Problem] --> B[GP Formulation]
    B --> C[Change of Variables]
    C --> D[Convex Problem]
    D --> E[Interior-Point Solver]
    E --> F[Optimal Design]
    style A fill:#ff9800
    style B fill:#ff9800
    style C fill:#2196f3
    style D fill:#4caf50
    style E fill:#9c27b0
    style F fill:#4caf50
</pre>
]

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### Challenges and Limitations

1. **Modeling Constraints**: Not all circuit behaviors can be easily expressed as posynomial constraints.
2. **Accuracy**: Simplified models used in GP may not capture all aspects of circuit behavior.
3. **Discrete Variables**: Standard GP doesn't handle discrete variables, which are common in digital design.

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### Recent Advances and Future Directions

1. **Signomial Programming**: Extensions to handle a broader class of functions.
2. **Integration with Machine Learning**: Using ML techniques to improve model accuracy and solve time.
3. **Mixed-Integer Geometric Programming**: Techniques to handle discrete variables in GP formulations.
4. **Multi-objective Optimization**: Developing methods to better handle multiple competing objectives.

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### 📚 Example: CMOS Inverter Sizing

Consider a simple CMOS inverter. A GP formulation might look like:

$$
\begin{array}{ll}
  \text{minimize} & \quad W_n + W_p \\
  \text{subject to} & \quad \frac{K}{W_n} + \frac{K}{W_p} \leq T_\text{delay} \\
  & \quad W_n \geq W_{min}, \quad W_p \geq W_{min}
\end{array}
$$

Where $Wn$ and $Wp$ are NMOS and PMOS widths, $K$ is a technology-dependent constant, and $T_\text{delay}$ is the target delay.

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### Conclusion

In conclusion, Geometric Programming has revolutionized many aspects of Electronic Design Automation by providing a powerful, efficient, and flexible optimization framework. As circuit designs become more complex and performance requirements more stringent, GP's role in EDA is likely to grow, driving innovations in both algorithm development and circuit design methodologies.

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# Q&A 🎤