A bi-objective problem doesn't quite optimize two objectives simultaneously: rather, it treats both objectives as equally important, and will give you a trade-off curve (the so-called Pareto front). At some points of that curve, you are making a trade-off in favor of objective1, at others, in favor of objective2. All points along the curve are feasible for the _same_ set of constraints, and this set of constraints does _not_ depend on either objective. I.e. the current value of the constraint fnts does not (explicitly) involve the value of either objective, only the independent variables.

A bi-level program is different: You really care about one objective, say f(x). And you want the optimum of f(x) over a set S which happens to be defined using another optimization (the lower level program): S = argmin{g(x,y): y \in Y}. For different values of x you get different values of S but this isn't a trade-off like in the bi-objective case: here you are seeking the optimum solution, and there's exactly one (though perhaps many optimizers)

Indeed we are talking about two separate entities of modeling frameworks. In fact, the two can be combined in a model where, for example, we can have several objectives at the so-called "upper level" of the bilevel program. Examples of such problems can be found in works such as:

http://www.optimization-online.org/DB_FILE/2010/03/2561.pdf

http://link.springer.com/article/10.1007%2Fs10957-009-9639-8#page-1

I think the area of bilevel optimization came up as natural modeling approaches in several areas of inquiry independently from each other.

In mechanical engineering, there are two types of variables, the main variables being those that define the structure (material choice, thickness of bars vs. void, etcetera), and the other the variables that define the quality of what has been built. We may call this the "response", which in the case of mechanical engineering are the functions of the stresses and displacements of the structure built once it has been subjected to external forces, etc. The variables that define the structure have priority, and are therefore said to be at the "upper level", while the "responses" in fact are results from a secondary optimization affected by parameters in the cost functions stemming from the actions of the "leaders" (at the upper level). We call these response variables the "lower level variables". 

In economics and transportation, the main variables are those that define the costs for the users/operators/travellers, while the responses are the actions that the followers will take once the prices (for example) are known. Here, "Stackelberg equilibria" and similar terms is classic. The lower level variables may be how many items of some type of good is bought, or how many will travel from A to B in a traffic network given link tolls. 

Dear Hamed,

The answer of Mr. Houssam Abbas was a clear and convincing explanation. Meanwhile, a similar question has been raised here:

https://www.researchgate.net/post/Can_I_swap_the_upper-level_problem_with_a_lower-level_one_in_Bilevel_Programming

I recommend you read its comments too.

Regards,

Morteza

http://mortezash.blogfa.com/

Can anyone provide a simple bilevel problem example and its necessary transformation to be programmed into a software (Phyton for example)?? Thanks in advance.

Can anyone provide a simple bilevel problem example and its necessary transformation to be programmed into a software (Phyton for example)?? Thanks in advance.

Can anyone provide a simple bilevel problem example and its necessary transformation to be programmed into a software (Phyton for example)?? Thanks in advance.

Dear Juan,

A simple bilevel example can be written like:

Min J(x) = x1^2 + a x2^2

s.t.

a < argmax(a - 2a^2)

This can be easily implemented in any optimization software.

However, a problem rises when the constraint an the first level optimization object are interdependent, and the solution is so tricky.