Two optimality tests for differentiable concave value functions in linear multi-objective programming problems (Q1095803)

The author outlines two tests which can be applied, under certain conditions, to solutions to linear multi-objective programming (MOP) problems Max \(c^ ix\) \((i=1,...,N)\), subject to \(Ax+I_ ms=b\), \(x\geq 0\), \(s\geq 0\), where \(c^ i\) is the (1\(\times n)\) gradient of the \(i^{th}\) objective function, x is the (n\(\times 1)\) vector of decision variables, A is the (m\(\times n)\) matrix of technological coefficients, \(I_ m\) is the (m\(\times m)\) identity matrix, s is the (m\(\times 1)\) vector of slack variables and b is the (m\(\times 1)\) vector of right hand side constants. The first test can be used to determine whether a given basic solution to the problem is optimal. The second test can be used to determine whether a given non-basic solution to the problem is optimal in a certain face of the set of feasible solutions of the problem.