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Quadratically constrained quadratic program

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Title: Quadratically constrained quadratic program  
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Quadratically constrained quadratic program

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 P0, … Pm are n-by-n matrices and xRn 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.

Contents

  • Hardness 1
  • Relaxation 2
    • Semidefinite programming 2.1
  • Example 3
  • Solvers and scripting (programming) languages 4
  • References 5
  • Further reading 6
    • In statistics 6.1
  • External links 7

Hardness

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.

Relaxation

There are two main relaxations of QCQP: using semidefinite programming (SDP), and using the reformulation-linearization technique (RLT).

Semidefinite programming

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.

Example

Max Cut is a problem in graph theory, which is NP-hard. 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 large-scale linear programs, quadratic programs and mixed-integer 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 cross-platform 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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