What is the appropriate optimization method (algorithm) to solve such problems (Linear mixed-integer)?

Question

I have this optimization problem:

In this problem, C_{i,k} is a matrix of binary values (i.e., 0 or 1) and w_i is a vector of integers, p_f is a probability, and \epsilon is a constant. I understand that the problem is a linear mixed-integer problem. But I'm confused about the method or algorithm I should use to solve the problem, and how can I go further by doing convexity analysis. Your inputs are appreciated. Thanks a lot.

Answer 1

This is a 0-1 knapsack problem. This problem can be solved using either dynamic programming or branch-and-bound algorithm. For branch and bound, you could select any variable z_k, solve two subproblems with z_k equals to either 0 or 1. Each subproblem has the exact structure as the original problem.