Quadratically constrained quadratic program explained

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} &minimize&&\tfrac12xTP0x+

T
q
0

x\\ &subjectto&&\tfrac12xTPix+

T
q
i

x+ri\leq0fori=1,...,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.

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 ∈ . 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). 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.

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

Solvers and scripting (programming) languages

NameBrief info
Artelys KnitroKnitro 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 XpressA 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
CPLEXPopular solver with an API for several programming languages. Free for academics.
MOSEKA solver for large scale optimization with API for several languages (C++, java, .net, Matlab and python)
TOMLABSupports 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 MathematicaAble to solve QCQP type of problems using functions like .

References

Further reading

In statistics

External links

Notes and References

  1. Kimizuka. Masaki. Kim. Sunyoung. Yamashita. Makoto. 2019. Solving pooling problems with time discretization by LP and SOCP relaxations and rescheduling methods. Journal of Global Optimization. en. 75. 3. 631–654. 10.1007/s10898-019-00795-w. 254701008 . 0925-5001.
  2. Kim. Sunyoung. Kojima. Masakazu. 2003. Exact Solutions of Some Nonconvex Quadratic Optimization Problems via SDP and SOCP Relaxations. Computational Optimization and Applications. 26. 2. 143–154. 10.1023/A:1025794313696. 1241391 .
  3. Burer. Samuel. Ye. Yinyu. 2019-02-04. Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs. Mathematical Programming. 181. 1–17. en. 10.1007/s10107-019-01367-2. 0025-5610. 1802.02688. 254143721 .