5.4.1. Modeling for Quadratically Constrained Programming¶
A Quadratically constrained program (referred to as QCP for simplicity) can be represented as follows:
- where
\(x \in \mathbb{R}^{n}\) is the decision variable,
\(l^c \in \mathbb{R}^{n}\) and \(u^c \in \mathbb{R}^{n}\) are the lower and upper bounds of \(x\),
\(c^f \in \mathbb{R}\) is the constant term in the objective function,
\(c \in \mathbb{R}^{n}\) is the linear coefficient in the objective function,
\(Q_0 \in \mathbb{R}^{n\times n}\) is the quadratic coefficient matrix in the objective function,
\(\{a_i \in \mathbb{R}^{n}\mid i=1,2,\cdots,m\}\) are the linear coefficient vectors of the constraints,
\(\{Q_i \in \mathbb{R}^{n\times n}\mid i=1,2,\cdots,m\}\) are the quadratic coefficient matrices of the constraints,
\(l^r \in \mathbb{R}^{m}\) and \(u^r \in \mathbb{R}^{m}\) are the lower and upper bounds of the constraints.
Note
Currently MindOpt supports solving convex quadratically constrained programming. Thus it requires the quadratic coefficient matrices in the objective \(Q_0\) and quadratic constraints \(\{Q_i \mid i=1,2,\cdots,m\}\) to be positive semi-definite. When users input quadratic constraints in the direction of \(\geq\), then it requires the quadratic matrices to be negative semi-definite.
We follow the steps below to use MindOpt :
Create an optimization model;
Input the optimization problem and set algorithm parameters;
Solve the optimization problem and retrieve the optimal solutions.
Note
MindOpt only stores the non-zero elements in the constraint matrices A and \(Q\). Therefore, we only need to input the positions and the values of the non-zero elements during the modeling process.
5.4.1.1. Example of Quadratically Constrained Programming¶
We consider the following Quadratically Constrained Programming problem:
We will provide example codes illustrating how MindOpt models and solves the above QCP problem in different programming languages.