How do I find elements with custom comparators with std::vector for large datasets?

#include <iostream>
#include <vector>
#include <algorithm>

struct CustomComparator {
    bool operator()(const int& a, const int& b) const {
        return a < b; // Define your custom comparison logic here
    }
};

int main() {
    std::vector<int> numbers = {5, 3, 8, 6, 2, 7, 4, 1};

    // Sort the vector using custom comparator
    std::sort(numbers.begin(), numbers.end(), CustomComparator());

    // Finding an element using binary search with custom comparator
    int target = 5;
    bool found = std::binary_search(numbers.begin(), numbers.end(), target, CustomComparator());

    if(found) {
        std::cout << "Element " << target << " found in the vector!" << std::endl;
    } else {
        std::cout << "Element " << target << " not found in the vector." << std::endl;
    }
    return 0;
}