![ A convex function c(x) defined on defined on [0, 1] ⊆ Rn can have many local optima,
but all must be global. This is illustrated in the following figure.](https://image.slidesharecdn.com/co-lec-4-210526055634/85/Combinatorial-optimization-CO-4-15-320.jpg)
Combinatorial optimization CO-4
- 1. Combinatorial Optimization CS-724 Lec-4: Date: 26-02-2021 Dr. Parikshit Saikia Assistant Professor Department of Computer Science and Engineering NIT Hamirpur (HP) India
- 2. Local and global optima Convex Set Convex Functions
- 5. Convex programming problem (CPP) The optimization problem of the form: Min f(x) subject to: gi (x) ≤ 0, i = 1, 2, 3,……..,m is called a CPP if f and gi (i = 1, 2, 3,…., m) are convex functions. Theorem: Let gi for each i = 1, 2, 3, …m be a convex function. Then S = {x Rn | gi (x) ≤ 0, i = 1, 2, 3, …m} is a convex set. i.e., the feasible set S in a CPP is convex.
- 7. How to determine if a functions is convex? Prove by definition Use properties Sum of convex functions is convex If 𝑓 𝑥 = σ𝑖 𝑤𝑖𝑓𝑖 𝑥 , 𝑤𝑖 ≥ 0, 𝑓𝑖 𝑥 convex, then 𝑓(𝑥) is convex Convexity is preserved under a linear transformation If 𝑓 𝑥 = 𝑔(𝐴𝑥 + 𝑏), 𝑔 convex, then 𝑓(𝑥) is convex If 𝑓 is a twice differentiable function of one variable, 𝑓 is convex on an interval 𝑎, 𝑏 ⊂ ℝ iff (if and only if) its second derivative 𝑓′′ 𝑥 ≥ 0 in 𝑎, 𝑏
- 8. If 𝑓 is a twice continuously differentiable function of 𝑛 variables, 𝑓 is convex on ℱ iff its Hessian matrix of second partial derivatives is positive semidefinite on the interior of ℱ 2/26/2021 Fei Fang 8 𝐻 is positive semidefinite in 𝑆 if ∀𝑥 ∈ 𝑆, ∀𝑧 ∈ ℝ𝑛, 𝑧𝑇𝐻(𝑥)𝑧 ≥ 0 𝐻 is positive semidefinite in ℝ𝑛 iff all eigenvalues of 𝐻 are non-negative Alternatively, prove 𝑧𝑇𝐻 𝑥 𝑧 = σ𝑖 𝑔𝑖 𝑥, 𝑧 2
- 13. Different formats of CPP Optimization Problem Condition for CPP Min f(x) subject to gi (x) ≤ 0, i = 1, 2, 3,...,m f and gi for all i are convex Max f(x) subject to gi (x) ≤ 0, i = 1, 2, 3,...,m f is concave and gi for all i are convex Min f(x) subject to gi (x) ≥ 0, i = 1, 2, 3,...,m f is convex and gi for all i are concave Max f(x) subject to gi (x) ≥ 0, i = 1, 2, 3,...,m f and gi for all i are concave
- 14. Theorem: In a CPP every point locally optimal with respect to the Euclidean distance neighborhood N is also globally optimal. Proof: Home work
- 15. A convex function c(x) defined on defined on [0, 1] ⊆ Rn can have many local optima, but all must be global. This is illustrated in the following figure.