I have a query regarding whether KKT is optimal with some linear inequality constraint and non-linear inequality constraint. For KKT to be optimal the inequality constraints must be convex.

We know that for the objective function to be concave with linear equality constraint (affine function) and non-linear inequality constraint (which must be convex), a feasible KKT point can be found which is the optimal [1, slide 35].

Ref [1]: R. Lusby, Karush-Kuhn-Tucker Conditions. 2015. [Online]. Available: http://www2.imm.dtu.dk/courses/02711/lecture3.pdf

For example: Find the optimal value of $x$ and $y$, where the variables $a,~b,~c,~d$ are the constants and greater than zero. \begin{gather} {max}(x^2+y^2)\\ s.t. i)~~ x+a \geq b, ~~ii)~~ y-c \geq d, ~~~~iii)~~ ax^2+bxy \leq c \end{gather}

  1. How will that linear inequality constraint be tackled in above condition?

  2. Is KKT is optimal for the above examples?


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