In standard convex programming, the objective function and each of the constraint inequalities are convex. in such case, if the KKT condition hold for a point, and Slater condition is also hold for the solution space, that point is global optimum. However what if one of the constraint isn't convex but the solution space is convex?

for example suppose that $g_1(\mathbf{x}),g_2(\mathbf{x})$ be two nonconvex functions and the set $S=\{\mathbf{x} | g_1(\mathbf{x})\le0 ,g_2(\mathbf{x})\le0\}$ is convex. Is the same result from KKT condition held? or is it held with a different constraint qualification? or not is not generally applicable in these case?

Actually I need a proof in if KKT condition (with/without new constraint qualification) is application or a counterexample in other cases.

In standard convex programming, the objective function and each of the constraint inequalities are convex. in such case, if the KKT condition hold for a point, and Slater condition is also hold for the solution space, that point is global optimum. However what if one of the constraint isn't convex but the solution space is convex?

for example suppose that $g_1(\mathbf{x}),g_2(\mathbf{x})$ be two nonconvex functions and the set $S=\{\mathbf{x} | g_1(\mathbf{x})\le0 ,g_2(\mathbf{x})\le0\}$ is convex. Is the same result from KKT condition held? or is it hold with a different constraint qualification? or is not generally applicable in these case?

Actually I need a proof which KKT condition (with/without new constraint qualification) is applicable otherwise a counterexample.

In standard convex programming, the objective function and each of the constraint inequalities are convex. in such case, if the KKT condition hold for a point, and Slater condition is also hold for the solution space, that point is global optimum. However what if one of the constraint isn't convex but the solution space is convex?

for example suppose that $g_1(\mathbf{x}),g_2(\mathbf{x})$ be two nonconvex functions and the set $S=\{\mathbf{x} | g_1(\mathbf{x})<0 ,g_2(\mathbf{x})<0\}$ is convex. Is the same result from KKT condition held? or is it held with a different constraint qualification? or not is not generally applicable in these case?

Actually I need a proof in if KKT condition (with/without new constraint qualification) is application or a counterexample in other cases.

In standard convex programming, the objective function and each of the constraint inequalities are convex. in such case, if the KKT condition hold for a point, and Slater condition is also hold for the solution space, that point is global optimum. However what if one of the constraint isn't convex but the solution space is convex?

for example suppose that $g_1(\mathbf{x}),g_2(\mathbf{x})$ be two nonconvex functions and the set $S=\{\mathbf{x} | g_1(\mathbf{x})\le0 ,g_2(\mathbf{x})\le0\}$ is convex. Is the same result from KKT condition held? or is it held with a different constraint qualification? or not is not generally applicable in these case?

Actually I need a proof in if KKT condition (with/without new constraint qualification) is application or a counterexample in other cases.

In standard convex programming, the objective function and each of the constraint inequalities are convex. in such case, if the KKT condition hold for a point, and Slater condition is also hold for the solution space, that point is global optimum. However what if one of the constraint isnt convex but the solution space is convex !

for example supposed that $g_1(\mathbf{x}),g_s(\mathbf{x})$ be two nonconvex functions and the set $S=\{\mathbf{x} | g_1(\mathbf{x})<0 ,g_2(\mathbf{x})<0\}$ is convex. Is the same result from KKT condition held  ? or is it held with a different constraint qualification  ? or not is not generally applicable in these case?

Actually I need a proof in if KKT condition (with/without new constraint qualification) is application or a counter example in other cases.

In standard convex programming, the objective function and each of the constraint inequalities are convex. in such case, if the KKT condition hold for a point, and Slater condition is also hold for the solution space, that point is global optimum. However what if one of the constraint isn't convex but the solution space is convex?

for example suppose that $g_1(\mathbf{x}),g_2(\mathbf{x})$ be two nonconvex functions and the set $S=\{\mathbf{x} | g_1(\mathbf{x})<0 ,g_2(\mathbf{x})<0\}$ is convex. Is the same result from KKT condition held? or is it held with a different constraint qualification? or not is not generally applicable in these case?

Actually I need a proof in if KKT condition (with/without new constraint qualification) is application or a counterexample in other cases.