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### Quasiconvex Optimization via Convex Feasibility Problems

#### Problem Formulation
- \( f_0(x) \leq t, \quad f_i(x) \leq 0, \quad i = 1, \ldots, m, \quad Ax = b \) \hfill (1)

#### Key Points
- For fixed \( t \), a convex feasibility problem in \( x \).
- If feasible, \( t \geq p^\star \); if infeasible, \( t \leq p^\star \).

#### Bisection Method for Quasiconvex Optimization

**Given:** \( l \leq p^\star, \, u \geq p^\star, \) tolerance \( \epsilon > 0 \).

**Repeat:**
1. \( t := (l + u)/2 \).
2. Solve the convex feasibility problem (1).
3. If (1) is feasible, set \( u := t \); else set \( l := t \).

**Until:** \( u - l \leq \epsilon \).

- Requires exactly \(\lceil \log_2((u - l)/\epsilon) \rceil\) iterations (where \( u, l \) are initial values).

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*Source: A. d’Aspremont. Convex Optimization M2.*
```
