Complexity Analysis of Primal-Dual Interior-Point Methods for Linear Optimization Based on a New Parametric Kernel Function with a Trigonometric Barrier Term

Table 1

Complexity results for the eligible kernel functions.

𝑖	The eligible kernel functions 𝜓𝑖(𝑡)	Large-update methods	Small-update methods	Reference
1	𝑡^2−12−log𝑡	𝑂𝑛𝑛log𝜀	𝑂√𝑛𝑛log𝜀	For example [1]
2	121𝑡−𝑡^2	𝑂𝑛^2/3𝑛log𝜀	𝑂√𝑛𝑛log𝜀	[3]
3	𝑡^2−12+𝑡^1−𝑞−1𝑞−1,𝑞>1	𝑂𝑞𝑛^(𝑞+1)/2𝑞𝑛log𝜀	𝑂𝑞^2√𝑛𝑛log𝜀	[3]
4	𝑡^2−12+𝑡^1−𝑞−1−𝑞(𝑞−1)𝑞−1𝑞(𝑡−1),𝑞>1	𝑂𝑞𝑛^(𝑞+1)/2𝑞𝑛log𝜀	𝑂𝑞^2√𝑛𝑛log𝜀	[2]
5	𝑡^2−12+𝑒^1/𝑡−𝑒𝑒	𝑂√𝑛(log𝑛)^2𝑛log𝜀	𝑂√𝑛𝑛log𝜀	[3]
6	𝑡^2−12−∫^𝑡1𝑒^1/𝜉−1𝑑𝜉	𝑂√𝑛(log𝑛)^2𝑛log𝜀	𝑂√𝑛𝑛log𝜀	[3]
7	𝑡^2−12+𝑒^𝑞(1/𝑡−1)−𝑞𝑞,𝑞≥1	𝑂𝑞√𝑛𝑛log𝜀	𝑂𝑞√𝑛𝑞𝑛log𝜀	[9]
8	𝑡^2−12−∫^𝑡1𝑒^𝑞(1/𝜉−1)𝑑𝜉,𝑞≥1	𝑂𝑞√𝑛𝑛log𝜀	𝑂𝑞√𝑛𝑞𝑛log𝜀	[3]
9	𝑡^2−12+6𝜋tan𝜋(1−𝑡)2+4𝑡	𝑂𝑛^3/4𝑛log𝜀	𝑂√𝑛𝑛log𝜀	[6]
10	𝑡^2−121−log(𝑡)+8tan^2𝜋(1−𝑡)2+4𝑡	𝑂𝑛^2/3𝑛log𝜀	𝑂√𝑛𝑛log𝜀	[7]
11	𝑡^2−12−∫^𝑡1𝑒^3(tan(𝜋2+2𝜉^)−1)𝑑𝜉	𝑂√𝑛(log𝑛)^2𝑛log𝜀	𝑂√𝑛𝑛log𝜀	[8]
12	𝑡𝑡−1+^1−𝑞−1𝑞−1,𝑞>1	𝑂𝑛𝑞𝑛log𝜀	𝑂𝑞^2√𝑛𝑛log𝜀	[3]
13	𝑡^𝑝+1−1+𝑡𝑝+1^1−𝑞−1[]𝑡𝑞−1,𝑡>0,𝑝∈0,1,𝑞>1𝑝+1−1[]𝑝+1−log𝑡,𝑡>0,𝑝∈0,1,𝑞=1	𝑂𝑞𝑛^(𝑝+𝑞)/(𝑞(1+𝑝))𝑛log𝜀	𝑂𝑞^2√𝑛𝑛log𝜀	[10]