# Lecture 12-2: Duality

Download the original slides: [CMSE382-Lec12_2.pdf](CMSE382-Lec12_2.pdf)

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## This Lecture

**Topics:**

- Strong duality in the convex case
- Dual for linear programming

**Announcements:**

- Homework 6 is due on Friday, April 17, 2026 at 11:59pm.

---

## Last Time

### Dual objective function

Consider the general model referred to as the primal model

$$
\begin{aligned}
& f^* = \text{min}  & & f(\mathbf{x}) \\
& \text{such that} & & g_i(\mathbf{x}) \leq 0, i=1,2,\ldots m, \\
&  & & h_j(\mathbf{x}) = 0, j=1,2,\ldots p, \\
&  & & \mathbf{x} \in X, \text{ where } X \subseteq \mathbb{R}^n,
\end{aligned}
$$

and $f, g_i,h_j$ are functions defined on $X$.

The Lagrangian of the problem is

$$
L(\mathbf{x},\boldsymbol{\lambda},\boldsymbol{\mu}) = f(\mathbf{x}) + \sum\limits_{i=1}^m{\lambda_i g_i(\mathbf{x})} + \sum\limits_{j=1}^p{\mu_j h_j(\mathbf{x})},
$$

The dual objective function $q: \mathbb{R}_+^m \times \mathbb{R}^p \to \mathbb{R} \cup \{-\infty\}$ is

$$
q(\boldsymbol{\lambda},\boldsymbol{\mu})=\min_{x\in X}{L(\mathbf{x},\boldsymbol{\lambda},\boldsymbol{\mu})},
$$

---

### Weak duality theorem

**Primal Problem**

$$
\begin{aligned}
& f^* = \text{min}  & & f(\mathbf{x}) \\
& \text{such that} & & g_i(\mathbf{x}) \leq 0, i=1,2,\ldots m, \\
&  & & h_j(\mathbf{x}) = 0, j=1,2,\ldots p, \\
&  & & \mathbf{x} \in X, \text{ where } X \subseteq \mathbb{R}^n,
\end{aligned}
$$

and $f, g_i,h_j$ are functions defined on $X$.

**Dual Problem**

$$
\begin{aligned}
& q^* = \text{max}  & & q(\boldsymbol{\lambda},\boldsymbol{\mu}) \\
& \text{such that} & &  (\boldsymbol{\lambda},\boldsymbol{\mu}) \in \text{dom}(q),
\end{aligned}
$$

where $\text{dom}(q)=\{(\boldsymbol{\lambda},\boldsymbol{\mu}) \in \mathbb{R}_{+}^m \times \mathbb{R}^p: q(\boldsymbol{\lambda},\boldsymbol{\mu}) > -\infty\}$, and $q(\boldsymbol{\lambda},\boldsymbol{\mu})=\min_{\mathbf{x}\in X}{L(\mathbf{x},\boldsymbol{\lambda},\boldsymbol{\mu})}$.

**Theorem (Weak duality theorem)**

Consider the primal problem and its dual. Then
$q^* \leq f^*$,
where $q^*, f^*$ are the optimal dual and primal values, respectively.

[Desmos example](https://www.desmos.com/3d/yppnizahux)

---

## Strong Duality

### Strong duality of convex problems with equality & inequality constraints

**Primal Problem (P)**

$$
\begin{aligned}
& f^* = \text{min}  & & f(\mathbf{x}) \\
& \text{such that} & & g_i(\mathbf{x})  \leq 0,  i=1,2,\ldots, m, \\
&                  & & h_j(\mathbf{x}) \le 0,  j=1,2,\ldots, p, \\
&                  & & s_k(\mathbf{x}) = 0,  k=1,2,\ldots, q, \\
&  & & \mathbf{x} \in X,
\end{aligned}
$$

**Dual Problem (D)**

$$
\begin{aligned}
& q^* = \text{max}  & & q(\boldsymbol{\lambda},\boldsymbol{\mu}) \\
& \text{such that} & &  (\boldsymbol{\lambda},\boldsymbol{\mu}) \in \text{dom}(q),
\end{aligned}
$$

where $\text{dom}(q)=\{(\boldsymbol{\lambda},\boldsymbol{\mu}) \in \mathbb{R}_{+}^m \times \mathbb{R}^p: q(\boldsymbol{\lambda},\boldsymbol{\mu}) > -\infty\}$, and $q(\boldsymbol{\lambda},\boldsymbol{\mu})=\min_{\mathbf{x}\in X}{L(\mathbf{x},\boldsymbol{\lambda},\boldsymbol{\mu})}$.

- For (P): $X$ is a convex set and $f,g_1,\ldots, g_m$ are convex functions over $X$. The functions $h_1,\ldots,h_p,s_1,\ldots,s_q$ are affine.

**Theorem (Strong duality under equality & inequality constraints)**

If the generalized Slater's condition is satisfied in (P) and $f^*$ has a finite optimal value, then the optimal value of (D) is attained, and the optimal values of the primal and dual problems are the same $f^*=q^*$.

---

### Duality gap

**Definition (Duality gap)**

The duality gap is the difference between the optimal primal value $f^*$ and the optimal dual value $q^*$ given by

$$
\Delta = f^* - q^*
$$

- $\Delta \geq 0$.
- $\Delta = 0$ if and only if strong duality holds.

---

### Dual for linear programming

**Primal**

$$
\begin{aligned}
& f^* = \text{min}  & & \mathbf{c}^{\top} \mathbf{x} \\
& \text{such that} & & A \mathbf{x} \leq \mathbf{b},
\end{aligned}
$$

**Dual**

$$
\begin{aligned}
& q^* = \text{max}  & & -\mathbf{b}^{\top} \boldsymbol{\lambda} \\
& \text{such that} & & A^{\top} \boldsymbol{\lambda}=-\mathbf{c},\\
& & & \boldsymbol{\lambda} \geq 0.
\end{aligned}
$$

**Strong duality holds**

If the primal problem is feasble (meaning the constraint set is not empty) and has a finite solution, then the optimal dual value is equal to the optimal primal value:

$$
q^*=f^*.
$$