版像 NJUAT 南京大学 人工智能学院 SCHODL OF ARTIFICIAL INTELUGENCE,NANJING UNIVERSITY Lecture 2.Convex Optimization Basics Advanced Optimization(Fall 2023) Peng Zhao zhaop@lamda.nju.edu.cn Nanjing University
Lecture 2. Convex Optimization Basics Peng Zhao zhaop@lamda.nju.edu.cn Nanjing University Advanced Optimization (Fall 2023)
(Constrained)Optimization Problem We adopt a minimization language min f(x) s.t.x∈X optimization variable x e Rd objective function:f:RdR feasible domain:C Rd Advanced Optimization(Fall 2023) Lecture 2.Convex Optimization Basics 2
Advanced Optimization (Fall 2023) Lecture 2. Convex Optimization Basics 2 (Constrained) Optimization Problem • We adopt a minimization language
Unconstrained Optimization The optimization variable is feasible over the whole Ra-space. min f(x) s.t.x∈Rd It is one of the most basic forms of mathematical optimization and serves as the foundations. ---"any optimization problem can be regarded as an unconstrained one" min :f(x) min h(x)f(x)+6x(x) s.t.x∈X s.t.x∈Rd barrier/indicator function 0, 6x(x)= x∈X, 00, x生X. Advanced Optimization(Fall 2023) Lecture 2.Convex Optimization Basics 3
Advanced Optimization (Fall 2023) Lecture 2. Convex Optimization Basics 3 Unconstrained Optimization • • It is one of the most basic forms of mathematical optimization and serves as the foundations. --- “any optimization problem can be regarded as an unconstrained one” barrier/indicator function
Convex Optimization This lecture focuses on the following simplified setting: Language:minimization problem Objective function:continuous and convex Feasible domain:a convex subset of Euclidean space ·What is a convex set? What is a convex function? ·How to minimize? Advanced Optimization(Fall 2023) Lecture 2.Convex Optimization Basics 4
Advanced Optimization (Fall 2023) Lecture 2. Convex Optimization Basics 4 Convex Optimization • This lecture focuses on the following simplified setting: • Language: minimization problem • Objective function: continuous and convex • Feasible domain: a convex subset of Euclidean space • What is a convex set? • What is a convex function? • How to minimize?
Outline Convex Set and Convex Function Convex Optimization Problem Optimality Condition ·Function Properties Advanced Optimization(Fall 2023) Lecture 2.Convex Optimization Basics 5
Advanced Optimization (Fall 2023) Lecture 2. Convex Optimization Basics 5 Outline • Convex Set and Convex Function • Convex Optimization Problem • Optimality Condition • Function Properties