CE 7426 Convex Optimization

## Elements of Convex Analysis - Part 1

• Mathematical Representation of Convex Sets
• Definition and Motivations
• Typical Convex Sets
• Convexity Preserving Operations
• Theories and Applications
• Proper Cone and Generalized Inequalities
• Minimum/Minimal Elements and Pareto Optimality
• Theorems

## Elements of Convex Analysis - Part 2

• Definition and Examples
• Convex Function
• Affine function is concave
• General Conditions of Convexity
• first-order condition of convexity
• second-order condition of convexity
• Related Topics
• Sublevel Set and Epigraph
• Jensen’s inequality
• Convexity Preserving Operations
• Composition with Affine Function
• Pointwise Maximum
• Pointwise Supremum
• Composition with Scalar and Vector Functions
• Minimization
• Perspective (Kullback-Leibler distance)
• Conjugate Function and Chernoff Bounds
• The Conjugata Function
• Chernoff’s Bound for Sum of i.i.d. Random Variables
• Convexity under Logarithm
• Log-Concave and Log-Convex Functions

## Optimization Problems

• Standard Form and Terminologies
• Problem Categorization
• Constrained and Unconstrained Problems
• Feasibility Problem: Problem without Objective
• Standard Form and Terminologies
• equality constrains define an affine
• feasible region is convex
• any locally optimal point is globally optimal
• Optimality Conditions
• First-Order Optimality Condition for Differentiable
• Typical Convex Optimization Problems
• Linear Program (LP)
• Bakery Problem
• Resource Problem
• Linear-Fractional Program
• Chebyshev Center of a Set
• Second-Order Cone Programming (SOCP)
• Robust Linear Programming
• Deterministic Robust LP as SOCP
• Stochastic Robust LP as SOCP
• Geometric Programming
• Wireless Communication as Geometric Programming
• Semidefinite Programming (SDP)
• SOCP as SDP
• Matrix Euclidean Norm Minimization
• Vector Optimization
• Optimal and Pareto Optimal Points
• Multi-Objective Optimization
• Scalarization
• Utility Fairness

## Lagrangian Duality

• Lagrangian Multiplier and Dual Function
• Lagrangian Multiplier
• Lagrangian and Dual Function: A Variational
• Interpretation Examples and Discussions
• Duality Theorems
• Weak and Strong Duality
• Constraint Qualifications and KKT Revisiting
• Examples and Discussions
• Perturbation and Sensitivity Analysis
• Relating Primal and Dual Functions
• Sensitivity Analysis