I want to find the projection of a vector called "a" on a closed and convex set with linear constraints. The set is in the following form: \begin{array}{ll} & Ax = b \\ & Bx \le d \\ &x \ge 0. \end{array} I know I need to solve the following problem to get the projection: \begin{array}{ll} \text{minimize} & || x-a||^2 \\ & Ax = b \\ & Bx \le d \\ &x \ge 0. \end{array} However, I am looking for an efficient way (instead of solving with commercial software like CPLEX) using the structure, convexity, and linearity of the set. I was wondering if anyone knows how to find the projection on this type of feasible region and space properties efficiently. Any resource that has addressed this issue will also help a lot. Note that I found this "Projection of $z$ onto the affine set $\{x\mid Ax = b\}$" here, but the KKT condition requires other conditions in my problem that lead to non-linear constraints. Thanks in advance for your help.
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You could use any quadratic program (QP) solver. For example, SQOPT. – copper.hat Dec 08 '18 at 02:28
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Thanks for your reply. CPLEX can solve quadratic objectives with the set of constraints, and I get the optimal solutions, but I am looking for efficient ways to solve this problem using mathematical properties like getting the solution by multiplying some matrices and doing similar things instead of optimizing using commercial software. – Mehrzad Dec 08 '18 at 23:22