I assume your looking for a mathematical solution, rather thanjust gong through the options one a a time and finding the cheapest (that may be quicker)

You want to minimise the total cost of training subject to a long list of constraints. Define the following variables:

flags to indicate if each training was purchased

the cost of each training

indicates if training i gives you skill j

You want to minimise the total cost, subject to having all skills learned.

subject to: Learning every skill

(skill 1 must be learned at least once)

(skill 2 must be learned at least once)

(skill 3 must be learned at least once)

...etc

Subjec to: Constraints for program conflicts:

(training 1 and 2 cant BOTH be taken)

(training 1 and 8 cant BOTH be taken)

(training 2 and 4 cant BOTH be taken)

(training 2 and 10 cant BOTH be taken)

Subject to: More attendance constraints:

(you can take training 1 at most 1 time)

(you can take training 2 at most 1 time)

(you can take training 2 at most 1 time)

...etc

Subject to: No negative answers

..etc

in principle, you can drop the "more attendance constraints" and get the same answer, because atttending the same program twice wont add any more skills but will increase the cost, so cant be the optimal solution

Hope that helps, dont ask me to solve it!!