If your feasible set is the set of all vectors whose components are integers,that is:

vector v belongs to feasible set S, v=[ v1 v2.... vn]' vi's are integer variables.

This set S is not a convex set, because if va and vb are vectors of S, any arbitrary convex combination of them does not belong to S.

For example:

va=[ 1 2 3 ]' , vb=[ 1 -1 0 ]', vc= (1/2) *va + (1/2) *vb =[ 1 1/2 3/2]' which does not belong to S

Integer optimization problems, also known as integer programming problems, are not considered convex optimization problems in general. Convex optimization problems involve optimizing a convex objective function over a convex set of feasible solutions, where the set of feasible solutions forms a convex set.

On the other hand, integer optimization problems involve optimizing a function subject to integer constraints on the decision variables. The requirement for the decision variables to take integer values introduces discrete or combinatorial aspects to the problem, which makes it fundamentally different from convex optimization.

The condition of an integer optimization problem is that one or more decision variables must take integer values. This constraint makes the problem more challenging to solve compared to continuous optimization problems, as it introduces additional complexity due to the discrete nature of the decision variables. The inclusion of integer constraints often results in a combinatorial search problem, as the solution space consists of a discrete set of possible combinations of integer values for the decision variables. Consequently, specialized solution methods, such as branch-and-bound algorithms or mixed-integer programming solvers, are typically employed to find optimal solutions for integer optimization problems.

In general, integer problems are not convex. First of all, due to nonconvexity of the feasible set. However, there are linear integer programming problems possessing properties similar to continuous linear problems. For example, classical transportation, assignment and many network flow problems. The reason is that all vertices of the feasible set have integer components. See more, e.g., in L.Wolsey, Integer Programming.

The feasible set of an integer optimization problem is countable or empty.

When the cardinality of the feasible set is precisely 1, then this set is convex, otherwise this set is not convex. Only in the case that the feasible set is convex, the problem can be reformulated , in principle, as a convex optimization problem. However, this is of no help in solving the problem.

An advise to the one, who posed the question: read a book about optimization theory, instead of posing this kind of questions . Too often I see at the Researchgate site badly phrased questions. It really pays to study decent books!