Convex optimization quadratic constraints

  • Are quadratic constraints convex?

    For the constraints, we normally require that the feasible set be a convex set, while the objective function should be convex.
    In general, the set of points (or vectors) satisfying a quadratic equality constraint may not be a convex set.Jan 6, 2016.

  • Are quadratic programs always convex?

    Quadratic Programming (QP) Problems
    The quadratic objective function may be convex -- which makes the problem easy to solve -- or non-convex, which makes it very difficult to solve.
    The "best" QPs have Hessians that are positive definite (in a minimization problem) or negative definite (in a maximization problem)..

  • Can Cplex handle quadratic constraints?

    CPLEX solves quadratic programs; that is, a model in which the constraints are linear, but the objective function can contain one or more quadratic terms.
    These problems are also known as QP.
    When such problems are convex, CPLEX normally solves them efficiently in polynomial time..

  • Is quadratic constraint convex?

    For the constraints, we normally require that the feasible set be a convex set, while the objective function should be convex.
    In general, the set of points (or vectors) satisfying a quadratic equality constraint may not be a convex set.Jan 6, 2016.

  • What is constraint quadratic programming?

    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..

  • Definition: An optimization problem for which the objective function, inequality, and equality constraints are linear is said to be a linear program.
    However, if the objective function is quadratic while the constraints are all linear, then the optimization problem is called a 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. where P0, …, Pm are n-by-n matrices and x ∈ Rn is the optimization variable.
  • The QCQP problem is known to be NP–hard in its general form; only in certain special cases can it be solved to global optimality in polynomial-time.
    Such cases are said to be convex in a hidden way, and the task of identifying them remains an active area of research.
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-  HardnessRelaxationSolvers and scripting References

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 constrai…

Relaxation

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

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 pos…

Further reading

• Albers C. J., Critchley F., Gower, J. C. (2011). "Quadratic Minimisation Problems in Statistics"(PDF). Journal of Multivariate Analysis. 102 (3): …

External links

• NEOS Optimization Guide: Quadratic Constrained Quadratic Programming

What is a general convex quadratically constrained quadratic optimization problem?

A general convex quadratically constrained quadratic optimization problem is where Q i ∈ S + n

This corresponds to minimizing a convex quadratic function over an intersection of convex quadratic sets such as ellipsoids or affine halfspaces

Convex optimization with linear equality constraints can also be solved using KKT matrix techniques if the objective function is a quadratic function (which generalizes to a variation of Newton's method, which works even if the point of initialization does not satisfy the constraints), but can also generally be solved by eliminating the equality constraints with linear algebra or solving the dual problem.

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