quadratically constrained quadratic program
{{Short description|Optimization problem in mathematics}}
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
:
& \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 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.
Hardness
A convex QCQP problem can be efficiently solved using an interior point method (in a polynomial time), typically requiring around 30-60 iterations to converge. Solving the general non-convex 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.
However, even for a nonconvex QCQP problem a local solution can generally be found with a nonconvex variant of the interior point method. In some cases (such as when solving nonlinear programming problems with a sequential QCQP approach) these local solutions are sufficiently good to be accepted.
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.{{Cite journal|last1=Kimizuka|first1=Masaki|last2=Kim|first2=Sunyoung|last3=Yamashita|first3=Makoto|date=2019|title=Solving pooling problems with time discretization by LP and SOCP relaxations and rescheduling methods|journal=Journal of Global Optimization|language=en|volume=75|issue=3|pages=631–654|doi=10.1007/s10898-019-00795-w|s2cid=254701008 |issn=0925-5001}}
Nonconvex QCQPs with non-positive off-diagonal elements can be exactly solved by the SDP or SOCP relaxations,{{Cite journal|last1=Kim|first1=Sunyoung|last2=Kojima|first2=Masakazu|date=2003|title=Exact Solutions of Some Nonconvex Quadratic Optimization Problems via SDP and SOCP Relaxations|journal=Computational Optimization and Applications|volume=26|issue=2|pages=143–154|doi=10.1023/A:1025794313696|s2cid=1241391 }} and there are polynomial-time-checkable sufficient conditions for SDP relaxations of general QCQPs to be exact.{{Cite journal|last1=Burer|first1=Samuel|last2=Ye|first2=Yinyu|date=2019-02-04|title=Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs|journal=Mathematical Programming|volume=181|pages=1–17|language=en|doi=10.1007/s10107-019-01367-2|issn=0025-5610|arxiv=1802.02688|s2cid=254143721 }} 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
- 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.
- QCQP is used to finely tune machine setting in high-precision applications such as photolithography.
Solvers and scripting (programming) languages
| class="wikitable" | |
| Name
!Brief info | |
|---|---|
| ALGLIB | ALGLIB, an open source/commercial numerical library, includes a QP solver supporting quadratic equality/inequality/range constraints, as well as other (conic) constraint types. |
| 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 {{mono|Minimize}}. |
| [https://clarabel.org/ clarabel] | Open source interior point numerical solver for convex optimization problems, supports second-order cone programming. |
References
{{Reflist}}
- {{cite book
| last = Boyd
| first = Stephen
|author2=Lieven Vandenberghe
| title = Convex Optimization
| publisher = Cambridge University Press
| year = 2004
| location = Cambridge
| url = https://web.stanford.edu/~boyd/cvxbook/
| isbn = 978-0-521-83378-3}}
Further reading
= In statistics =
- {{cite journal |author=Albers C. J., Critchley F., Gower, J. C. |title=Quadratic Minimisation Problems in Statistics |journal=Journal of Multivariate Analysis |volume=102 |issue=3 |pages=698–713 |year=2011 |doi=10.1016/j.jmva.2009.12.018 |url=https://pure.rug.nl/ws/files/111582994/1_s2.0_S0047259X09002498_main.pdf |hdl=11370/6295bde7-4de1-48c2-a30b-055eff924f3e |hdl-access=free }}
External links
- [http://neos-guide.org/content/quadratic-constrained-quadratic-programming NEOS Optimization Guide: Quadratic Constrained Quadratic Programming]