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

\begin{align} & \text{minimize} && \tfrac12 x^\mathrm{T} P_0 x + q_0^\mathrm{T} x \\ & \text{subject to} && \tfrac12 x^\mathrm{T} P_i x + q_i^\mathrm{T} x + r_i \leq 0 \quad \text{for } i = 1,\dots,m , \\ &&& Ax = b, \end{align}
where P_{0}, … P_{m} are nbyn matrices and x ∈ R^{n} is the optimization variable.
If P_{0}, … P_{m} are all positive semidefinite, then the problem is convex. If these matrices are neither positive nor negative semidefinite, the problem is nonconvex. If P_{1}, … P_{m} are all zero, then the constraints are in fact linear and the problem is a quadratic program.
Contents

Hardness 1

Relaxation 2

Semidefinite programming 2.1

Example 3

Solvers and scripting (programming) languages 4

References 5

Further reading 6

External links 7
Hardness
Solving the general case is an NPhard problem. To see this, note that the two constraints x_{1}(x_{1} − 1) ≤ 0 and x_{1}(x_{1} − 1) ≥ 0 are equivalent to the constraint x_{1}(x_{1} − 1) = 0, which is in turn equivalent to the constraint x_{1} ∈ {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 NPhard in general, QCQP is also NPhard.
Relaxation
There are two main relaxations of QCQP: using semidefinite programming (SDP), and using the reformulationlinearization technique (RLT).
Semidefinite programming
When P_{0}, … P_{m} are all positivedefinite matrices, the problem is convex and can be readily solved using interior point methods, as done with semidefinite programming.
Example
Max Cut is a problem in graph theory, which is NPhard. Given a graph, the problem is to divide the vertices in two sets, so that as many edges as possible go from one set to the other. Max Cut can be formulated as a QCQP, and SDP relaxation of the dual provides good lower bounds.
Solvers and scripting (programming) languages
Name

Brief info

AMPL


CPLEX

Popular solver with an API for several programming languages. Free for academics.

Gurobi

Solver with parallel algorithms for largescale linear programs, quadratic programs and mixedinteger programs. Free for academic use.

JOptimizer

Java library for convex optimization (open source)

MOSEK

A solver for large scale optimization with API for several languages (C++,java,.net, Matlab and python)

OpenOpt

universal crossplatform numerical optimization framework, see its QCQP page and other problems involved. Uses NumPy arrays and SciPy sparse matrices.

TOMLAB

Supports global optimization, integer programming, all types of least squares, linear, quadratic and unconstrained programming for MATLAB. TOMLAB supports solvers like Gurobi, CPLEX, SNOPT and KNITRO.

References
Further reading
In statistics
External links

NEOS Optimization Guide: Quadratic Constrained Quadratic Programming
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