# Difference between Chance constraints and logical constraints

A logical constraint combines linear constraints using logical operators, such as logical-and, logical-or, negation (that is, not), conditional statements (that is, if ... then ...) to express complex relations between linear constraints.

About chance constraints, it is desired to specify that a certain constraint be satisfied with a given probability. $$$$P\left[\sum_{j} a_{j} x_{j} \leq b\right] \geq \beta$$$$ According to the above definitions, Is there any difference between Logical constraints and chance constraints?

• Out of curiosity, can you explain why/how these two types of constraints are related? – David M. Aug 3 '19 at 2:57
• @DavidM., some days ago, I saw a comment on the forum of one of the optimization solvers that, I can use logical constraints to deal with chance constraints. – A.Omidi Aug 3 '19 at 4:24
• Can you provide a link to or extract from the comment? – Mark L. Stone Aug 3 '19 at 11:50

Logical constraints do not involve probability, except perhaps for the implicit probability of one or zero.

Chance constraints specify conditions (constraints) which must hold with a(t least) specified probability, which generally would not be one or zero.

Chance constraints could include logical conditions, and potentially even be specified in terms of holding with specified conditional probability, conditional on specified conditions.

Also note that neither logical constraints nor chance constraints need be linear.

• you are so fast... it took 20 minutes for me to find some useful links. – Oguz Toragay Aug 2 '19 at 23:44

These two types of constraint are totally different in terms of their applications in modeling. In fact, the way of using these constraint types (based on your modeling approach) end up in two totally distinct problems each of which can be solved different solution approaches. In the following, I will try to explain where we need to implement each of the constraint types:

• The most direct way to treat stochastic data uncertainty in the context of uncertain Linear Optimization is offered by an old concept (going back to 50s) of chance constraints [source]. So you can use the chance constraint to deal with uncertainty in the problem and some of the possible approaches to solving this kind of problems are, Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR).

• Logical constraints, on the other hand, do not enclose uncertainty. Most of the time logical statements in the problem can be explained and modeled by using constraints that shadow forth the logical situations in the problem.Here is a good explanation of how to use binary variables to model the logical situations in the form of logical constraints. To solve the problems with logical constraints most of the solvers can be easily used. This link, includes the details and examples of modeling logical situation in the constraints in one of the commercial modeling languages.

• @Mark L. Stone and Oguz Toragay, thanks so much for your comments. – A.Omidi Aug 3 '19 at 4:25