<p>Convex optimization problems in the presence of regularity, inherit the benefits of strong duality and tractability in-order to obtain global optimal solution(s). On the other hand, typical non-convex optimization problems lack the two interrelated key characteristics, which generally results in computationally expensive solution methods. In this work, we present Non-Convex Optimization Problems (NCOPs) that minimize the sum of concave and affine functions, where the concave function is a ridge type function. The three characteristics of the non-convex problems that pave the path for the efficient solution approach are: regularity of the feasible region, concave minimization over polytope, and linear KKT subsystem. In the current work, sufficient conditions for the existence of a linear KKT subsystem are proposed. Six synthetic test instances are used to illustrate the performance of the proposed approaches. The results indicate that the proposed approaches are efficient (polynomial time) in solving the NCOPs that have the three highlighted characteristics.</p>