Advanced Mathematics for Computer Science HNCST409 KTU BTech Honors 2024 Scheme- Dr Binu V P

  Question Show that the function f ( x ) = e x f(x) = e^{x} is a convex function on R \mathbb{R} . Solution To show that f ( x ) = e x f(x) = e^x is convex, we use the second derivative test for convexity . Step 1: Compute the First Derivative f ′ ( x ) = d d x ( e x ) = e x f'(x) = \frac{d}{dx}(e^x) = e^x Step 2: Compute the Second Derivative f ′ ′ ( x ) = d 2 d x 2 ( e x ) = e x f''(x) = \frac{d^2}{dx^2}(e^x) = e^x Step 3: Apply the Convexity Test Since f ′ ′ ( x ) = e x > 0 ∀ x ∈ R , f''(x) = e^x > 0 \quad \forall x \in \mathbb{R}, the second derivative is positive everywhere . Conclusion A function whose second derivative is non-negative on an interval is convex on that interval. Hence, f ( x ) = e x  is a convex function on  R . \boxed{ f(x) = e^x \text{ is a convex function on } \mathbb{R}. } ​ (Optional) Remark for Deeper Understanding The exponential function satisfies Jensen’s inequality: e λ x 1 + ( 1 − λ ) x 2 ≤ λ...

  Applications of Convex Sets Convex sets play a crucial role in many areas of mathematics, engineering, economics, and data science. Their simple geometric structure leads to strong theoretical guarantees and efficient algorithms. 1. Optimization Convex sets form the foundation of convex optimization problems. The feasible region defined by convex constraints is a convex set. When both the objective function and the constraint set are convex: Every local minimum is a global minimum The solution, if it exists, is often unique Efficient algorithms such as gradient descent and interior-point methods can be applied Examples Linear programming Quadratic programming Support Vector Machines (SVMs) LASSO and Ridge regression 2. Machine Learning Many classical machine learning models rely on convex sets. Parameter constraints (e.g., norm balls for regularization) are convex sets Feasible hypothesis spaces are often convex Convex loss fu...

  Convex Hull 1. Intuitive Idea The convex hull of a set of points is the smallest convex set that contains all those points. 🔹 Intuition : Imagine driving nails at given points on a board and stretching a rubber band around them. When released, the rubber band forms the convex hull . 2. Formal Definition Let S ⊆ R n S \subseteq \mathbb{R}^n  be any set. The convex hull of S S , denoted by conv ⁡ ( S ) , \operatorname{conv}(S), is defined as the set of all convex combinations of points in S S S . That is, conv ⁡ ( S ) = { ∑ i = 1 k λ i x i    |    x i ∈ S ,    λ i ≥ 0 ,    ∑ i = 1 k λ i = 1 ,    k ∈ N } \boxed{ \operatorname{conv}(S) = \left\{ \sum_{i=1}^k \lambda_i x_i \;\middle|\; x_i \in S,\; \lambda_i \ge 0,\; \sum_{i=1}^k \lambda_i = 1,\; k \in \mathbb{N} \right\} } ​ 3. Convex Combination (Key Concept) A convex combination of points x 1 , x 2 , … , x k x_1, x_2, \dots, x_k  is a weighted average: λ 1 x 1 + λ 2 x 2 + ⋯ + λ k x k \lambda_1 x_1 + ...