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In mathematical optimization, a quadratically constrained quadratic program (QCQP) is an optimization problem in which both the objective function and the constraints are quadratic functions. It has the form
where P0, ..., Pm are n-by-n matrices and x ∈ Rn is the optimization variable.
If P0, ..., Pm are all positive semidefinite, then the problem is convex. If these matrices are neither positive nor negative semidefinite, the problem is non-convex. If P1, ... ,Pm are all zero, then the constraints are in fact linear and the problem is a quadratic program.
Solving the general case is an NP-hard problem. To see this, note that the two constraints x1(x1 − 1) ≤ 0 and x1(x1 − 1) ≥ 0 are equivalent to the constraint x1(x1 − 1) = 0, which is in turn equivalent to the constraint x1 ∈ {0, 1}. Hence, any 0–1 integer program (in which all variables have to be either 0 or 1) can be formulated as a quadratically constrained quadratic program. Since 0–1 integer programming is NP-hard in general, QCQP is also NP-hard.
There are two main relaxations of QCQP: using semidefinite programming (SDP), and using the reformulation-linearization technique (RLT). For some classes of QCQP problems (precisely, QCQPs with zero diagonal elements in the data matrices), second-order cone programming (SOCP) and linear programming (LP) relaxations providing the same objective value as the SDP relaxation are available.[1]
Nonconvex QCQPs with non-positive off-diagonal elements can be exactly solved by the SDP or SOCP relaxations,[2] and there are polynomial-time-checkable sufficient conditions for SDP relaxations of general QCQPs to be exact.[3] Moreover, it was shown that a class of random general QCQPs has exact semidefinite relaxations with high probability as long as the number of constraints grows no faster than a fixed polynomial in the number of variables.[3]
When P0, ..., Pm are all positive-definite matrices, the problem is convex and can be readily solved using interior point methods, as done with semidefinite programming.
Name | Brief info |
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Artelys Knitro | Knitro is a solver specialized in nonlinear optimization, but also solves linear programming problems, quadratic programming problems, second-order cone programming, systems of nonlinear equations, and problems with equilibrium constraints. |
FICO Xpress | A commercial optimization solver for linear programming, non-linear programming, mixed integer linear programming, convex quadratic programming, convex quadratically constrained quadratic programming, second-order cone programming and their mixed integer counterparts. |
AMPL | |
CPLEX | Popular solver with an API for several programming languages. Free for academics. |
MOSEK | A solver for large scale optimization with API for several languages (C++, java, .net, Matlab and python) |
TOMLAB | Supports global optimization, integer programming, all types of least squares, linear, quadratic and unconstrained programming for MATLAB. TOMLAB supports solvers like CPLEX, SNOPT and KNITRO. |
Wolfram Mathematica | Able to solve QCQP type of problems using functions like Minimize. |
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