1. Mathematical Derivation: Is \(f(x,y) = x^2 + y^2\) Convex?
Gradient
∇f(x,y) = [ 2x, 2y ]แต
Hessian
∇²f(x,y) = [ 2 0
0 2 ]
Check Positive Semidefiniteness
For any vector z = [z1, z2]แต:
zแต ∇²f z = 2z₁² + 2z₂² ≥ 0
Thus, the Hessian is positive definite, so:
→ \(f(x,y)\) is strictly and strongly convex.
2. Does “Convex” Mean the Minimum is Zero?
No. Convexity refers to the shape of the function, not the value of the minimum.
A convex function can have any minimum value.
For the special case of \(x^2 + y^2\):
∂f/∂x = 2x = 0 → x = 0
∂f/∂y = 2y = 0 → y = 0
f(0,0) = 0
But zero is not special — it is just the minimum of this specific function.
3. Convex Optimization Techniques Used in Wireless Communication
1. Semidefinite Relaxation (SDR)
- MIMO/MISO beamforming
- SINR constraints
- Power minimization
2. Second-Order Cone Programming (SOCP)
- Beamforming
- Robust power control
- Relay selection
3. Geometric Programming (GP)
- Power control
- Energy-efficiency optimization
- Interference management
4. Water-Filling Algorithms
- OFDM power allocation
- Channel capacity maximization
5. Lagrangian Dual Decomposition
- Resource allocation
- Multi-cell coordination
6. Interior-Point Methods
- LP, QP, SOCP, SDP solvers
- Used via CVX, CVXPY, MOSEK
7. Alternating Convex Optimization (ACO)
- RIS optimization
- NOMA systems
- Joint beamforming & power control