# Strong duality strictly convex quadratic problem

Assume we have this strictly convex quadratic programming:

$$f(x) = x^\top A x + b^\top x,$$ $$Ax \leq b$$ $$0 \leq x \leq 1$$

Where $A$ is symmetric and positive definite, and the feasible set is nonempty. Does strong duality and Slater's condition holds in this case.

• you said the feasible set is nonempty, so I do not understand your first question (feasibility means that the optimal value is finite too) – LinAlg Jun 30 '18 at 12:25
• you mean the second question, right, since the first one is about the duality? I removed the second question, how about the duality? – Faroq AL-Tam Jun 30 '18 at 12:30
• Slater's condition requires a strictly feasible solution. If your linear equalities are such that (e.g.) there is a unique feasible solution then Slater's condition wouldn't be satisfied. Depending on your constraints, other constraint qualifications, such as the linear independence constraint qualification (LICQ), might hold. – Brian Borchers Jun 30 '18 at 14:19
• @Brian Borchers All the constraints are linear, that is a constraint qualification. Slater's condition only pertains to nonlinear constraints, so you could say it is trivially satisfied in this case. – Mark L. Stone Jun 30 '18 at 15:33
• @MarkL.Stone yes, but I was hoping that the OP could find this out for themselves... – Brian Borchers Jun 30 '18 at 15:56

• Thank you, Does this proof hold if we also have additional equality constraints $Cx = 1$. – Faroq AL-Tam Jun 30 '18 at 14:45