Quadratic constrained quadratic programming

Author: Jialiang Wang (jw2697), Jiaxin Zhang (jz2289), Wenke Du (wd275), Yufan Zhu (yz2899), David Berroa (deb336) (ChemE 6800 Fall 2024)

Stewards: Nathan Preuss, Wei-Han Chen, Tianqi Xiao, Guoqing Hu

Introduction

Quadratic programming (QP) is one of the oldest topics in the field of optimization that researchers have studied in the twentieth century. The basic QP, where the objective function is quadratic and constraints are linear, paved the way for other forms, such as QCQPs, which also have quadratic constraints (McCarl,Moskowitz et al,1977).

A Quadratically Constrained Quadratic Program (QCQP) can be defined as an optimization problem where the objective function and the constraints are quadratic. It emerged as optimisation theory grew to address more realistic, complex problems of non-linear objectives and constraints. In particular, the issue involves optimizing (or minimizing) a convex quadratic function of decision variables with quadratic constraints. This class of problems is well suited to finance (Zenios,1993), engineering, machine learning, and agriculture because it is easier to model the relationship between variables using quadratic functions.

The desire to study QCQPs stems from the fact that they can be used to model practical optimization problems that involve stochasticity in risk, resources, production, and decision-making. For example, in agriculture, using QCQPs can be useful in determining the best crop to grow based on the expected profits and the uncertainties of price changes and unfavourable weather conditions (Floudas,1995).
In finance, the QCQPs are applied in the portfolio's construction to maximize the portfolio's expected returns and the covariance between the assets. It is crucial to comprehend QCQPs and the ways to solve them, such as KKT (Karush-Kuhn Tucker) conditions and SDP (Semidefinite Programming) relaxations, to solve problems that linear models cannot effectively solve (Bao & Sahinidis,2011).


Name	Brief info
KKT(Karush-Kuhn-Tucker)	KKT is a mathematical optimization method used to solve constrained optimization problems(Ghojogh, Karray et al,2021). It builds upon the method of Lagrange multipliers by introducing necessary conditions for optimality that incorporate primal and dual variables. The KKT conditions include stationarity, primal feasibility, dual feasibility, and complementary slackness, making it particularly effective for solving problems with nonlinear constraints.
SDP (Semidefinite Programming)	SDP reformulates a QCQP problem into a semidefinite programming relaxation (Freund,2004). By “lifting” the problem to a higher-dimensional space and applying SDP relaxation, this approach provides a tractable way to solve or approximate solutions to non-convex QCQP problems. It is widely used in areas where global optimization or approximations to non-convex issues are necessary.

In general, analyzing QCQPs is important in order to apply knowledge-based decision-making and enhance the performance and stability of optimization methods in different fields (Zenios,1993).

Algorithm Discussion

KKT： What is KKT +formulation

SDP: What is SDP +formulation

Selected Solvers:


Name	Brief Info

Numerical Example

Quadratically Constrained Quadratic Program (QCQP) always has the form:

${\begin{array}{ll}\operatorname {minimize} &{\frac {1}{2}}x^{\mathrm {T} }P_{0}x+q_{0}^{\mathrm {T} }x\\{\text{ subject to }}&{\frac {1}{2}}x^{\mathrm {T} }P_{i}x+q_{i}^{\mathrm {T} }x+r_{i}\leq 0\quad {\text{ for }}i=1,\ldots ,m\\&Ax=b,\end{array}}$

where $P_{0},\ldots ,P_{m}$ are n-by-n matrices and x $\in \mathbf {R} ^{n}$ is the optimization variable. If $P_{0},\ldots ,P_{m}$ are all positive semidefinite, then the problem is convex. If these matrices are neither positive nor negative semidefinite, the problem is non-convex. If where $P_{0},\ldots ,P_{m}$ are all zero, then the constraints are in fact linear and the problem is a quadratic program.

Example 1: KKT Approach

Considering the following numerical example:

${\begin{aligned}{\text{minimize}}\quad &f_{0}(x)=(x_{1}-2)^{2}+x_{2}^{2}\\{\text{subject to}}\quad &f_{1}(x)=x_{1}^{2}+x_{2}^{2}-1\leq 0,\\&f_{2}(x)=(x_{1}-1)^{2}+x_{2}^{2}-1\leq 0.\end{aligned}}$

Steps:

Formulate the Lagrangian and computed the Gradients
Applied the Stationarity Conditions
Determined the active constraints using complementary slackness

1. 1 Lagrangian Formulation:

The Lagrangian formulation in optimization is a mathematical framework used to solve constrained optimization problems by incorporating both the objective function and the constraints into a single scalar function, called the Lagrangian $L$ This formulation introduces Lagrange multipliers $\lambda _{i}$ for each constraint, enabling the transformation of a constrained optimization problem into an unconstrained one as follows:

$L(x,\lambda ,\nu )=f_{0}(x)+\sum _{i=1}^{m}\lambda _{i}f_{i}(x)+\nu ^{T}(Ax-b)$

where:

$f_{0}(x)$ is the objective function to be minimized, $f_{i}(x)\leq 0$ are the inequality constraints

$Ax=b$ represents the equality constraints, $\quad \lambda _{i}\geq 0$ are the Lagrange multipliers associated with the inequality constraints, $\quad \nu$ is the Lagrange multiplier vector for the equality constraints.

Here it is:

$L(x,\lambda _{1},\lambda _{2})=(x_{1}-2)^{2}+x_{2}^{2}+\lambda _{1}(x_{1}^{2}+x_{2}^{2}-1)+\lambda _{2}\left((x_{1}-1)^{2}+x_{2}^{2}-1\right).$

For each constraint,

the complementary slackness is $\lambda _{i}\geq 0,\quad \lambda _{i}f_{i}(x)=0,\quad {\text{ for }}i=1,2$
the primal feasibility is $f_{i}(x)\leq 0\quad {\text{ for }}i=1,2$ .

The results for gradient computation are:

the partial derivatives with respect to $x_{1}$ : ${\frac {\partial L}{\partial x_{1}}}=2\left(x_{1}-2\right)+2\lambda _{1}x_{1}+2\lambda _{2}\left(x_{1}-1\right)$
the partial derivatives with respect to $x_{2}$ : ${\frac {\partial L}{\partial x_{2}}}=2x_{2}+2\lambda _{1}x_{2}+2\lambda _{2}x_{2}.$

1. 2 Stationarity Condition Application:

1.2.1 Setting the results to zero

$2(x_{1}-2)+2\lambda _{1}x_{1}+2\lambda _{2}(x_{1}-1)=0$
$2x_{2}+2\lambda _{1}x_{2}+2\lambda _{2}x_{2}=0$ $2x_{2}+2\lambda _{1}x_{2}+2\lambda _{2}x_{2}=0$
- since $x_{2}(1+\lambda _{1}+\lambda _{2})=0$ and $\lambda _{i}\geq 0$ for $i=1,2$ , so $x_{2}=0.$
with constrains $x_{1}\in [0,1].$

1.2.2 Substituting $x_{2}=0$ into the constraints

${\begin{aligned}x_{1}^{2}-1&\leq 0\quad \Rightarrow \quad x_{1}\in [-1,1],\\(x_{1}-1)^{2}-1&\leq 0\quad \Rightarrow \quad x_{1}\in [0,2].\end{aligned}}$

1.2.3 Problem Solving

Substitute $x_{2}=0$ $x_{2}=0$ into Equation (1): $(x_{1}-2)+\lambda _{1}x_{1}+\lambda _{2}(x_{1}-1)=0.$ $(x_{1}-2)+\lambda _{1}x_{1}+\lambda _{2}(x_{1}-1)=0.$
- Assume $\lambda _{1}>0$ (since Constraint 1 is active): $x_{1}^{2}-1=0\quad \Rightarrow \quad x_{1}=\pm 1.$
But from the feasible range, $x_{1}=1$ $x_{1}=1$
- Substitute $x_{1}=1$ $x_{1}=1$ into the equation: $\lambda _{1}=1.$ $\lambda _{1}=1.$
  - This is acceptable.
Assume $\lambda _{2}=0$ because Constraint 2 is not active at $x_{1}=1$ .

1. 3 Verification

1.3.1 Complementary Slackness Verification

Constraint 1: $\lambda _{1}(x_{1}^{2}-1)=1\times (1-1)=0.$
Constraint 2: $\lambda _{2}\left((x_{1}-1)^{2}+x_{2}^{2}-1\right)=0\times (-1)=0.$

1.3.2 Primal Feasibility Verification

Constraint 1: $x_{1}^{2}-1=1-1=0\leq 0$
Constraint 2: $(x_{1}-1)^{2}+x_{2}^{2}-1=-1\leq 0.$

1. 4 Conclusion

Optimal Solution: $x_{1}^{*}=1,\quad x_{2}^{*}=0.$
Minimum Objective Value : $f_{0}^{*}(x)=(1-2)^{2}+0=1.$

Example 2: SDP- Based QCQP

Consider the following Quadratically Constrained Quadratic Programming (QCQP) problem:

${\begin{aligned}{\text{minimize}}\quad &f_{0}(x)=x_{1}^{2}+x_{2}^{2}\\{\text{subject to}}\quad &f_{1}(x)=x_{1}^{2}+x_{2}^{2}-2\leq 0,\\&f_{2}(x)=-x_{1}x_{2}+1\leq 0.\end{aligned}}$ Interpretation:

The objective $f_{0}(x)=x_{1}^{2}+x_{2}^{2}$ is the squared distance from the origin.A point is found in the feasible region that is as close as possible to the origin.

The constraint $f_{1}(x)=x_{1}^{2}+x_{2}^{2}-2\leq 0$ restricts $(x_{1},x_{2})$ to lie inside or on a circle of radius ${\sqrt {2}}$ . The constraint $f_{2}(x)=-x_{1}x_{2}+1\leq 0\implies x_{1}x_{2}\geq 1$ defines a hyperbolic region. To satisfy $x_{1}x_{2}\geq 1$ , both variables must be sufficiently large in magnitude and have the same sign.

Step 1: Lifting and Reformulation

Introduce the lifted variable:

$x={\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}},\quad X=xx^{T}={\begin{pmatrix}x_{1}^{2}&x_{1}x_{2}\\x_{1}x_{2}&x_{2}^{2}\end{pmatrix}}.$ If $X=xx^{T}$ , then $X\succeq 0$ (positive semidefinite) and $X$ is rank-1.

Rewrite the objective and constraints in terms of $X$ :

Objective: $x_{1}^{2}+x_{2}^{2}=\langle I,X\rangle$ , where $I$ is the 2x2 identity matrix.

Constraint 1: $x_{1}^{2}+x_{2}^{2}-2\leq 0\implies \langle I,X\rangle -2\leq 0.$

Constraint 2: $-x_{1}x_{2}+1\leq 0\implies X_{12}\geq 1.$

Step 2: SDP Relaxation

The original QCQP is non-convex due to the rank-1 condition on $X$ . Relax the rank constraint and consider only $X\succeq 0$ :

${\begin{aligned}{\text{minimize}}\quad &\langle I,X\rangle \\{\text{subject to}}\quad &\langle I,X\rangle -2\leq 0,\\&X_{12}\geq 1,\\&X\succeq 0.\end{aligned}}$ This is now a Semidefinite Program (SDP), which is convex and can be solved using standard SDP solvers.

Step 3: Solving the SDP and Recovering the Solution

Solving the SDP, the feasible solution $X^{*}$ found that achieves the minimum:

$X^{*}={\begin{pmatrix}1&1\\1&1\end{pmatrix}}.$ Check that $X^{*}$ is rank-1:

$X^{*}={\begin{pmatrix}1\\1\end{pmatrix}}{\begin{pmatrix}1&1\end{pmatrix}}=x^{*}(x^{*})^{T},$ with $x^{*}=(1,1)$ .

Thus, from ${\text{Thus, from }}X^{*},{\text{ we recover the solution }}x^{*}=(1,1){\text{ for the original QCQP.}}$

Check feasibility:

$x_{1}^{2}+x_{2}^{2}=1+1=2\implies f_{1}(x^{*})=0\leq 0.$ $x_{1}x_{2}=1\implies f_{2}(x^{*})=-1+1=0\leq 0.$ All constraints are satisfied.

Step 4: Optimal Value

The optimal objective value is: $f_{0}^{*}(x)=x_{1}^{*2}+x_{2}^{*2}=1+1=2.$

Conclusion

The SDP relaxation not only provides a lower bound but also recovers the global optimum for this particular QCQP. The optimal solution is $(x_{1}^{,}x_{2}^{)}=(1,1)$ with an objective value of $2$ .

This example demonstrates how an SDP relaxation can be used effectively to solve a non-convex QCQP and, in some cases, recover the exact optimal solution.

Application

Conclusion

In conclusion, Quadratically Constrained Quadratic Programs (QCQPs) are a significant class of optimization problems extending quadratic programming by incorporating quadratic constraints (Bao,Sahinidis,2011). These problems are essential for modeling complex real-world scenarios where both the objective function and the constraints are quadratic. QCQPs are widely applicable in areas such as agriculture, finance, production planning, and machine learning, where they help optimize decisions by balancing competing factors such as profitability, risk, and resource constraints.

The study and solution of QCQPs are critical due to their ability to capture complex relationships and non-linearities, offering a more realistic representation of many practical problems than simpler linear models. Techniques such as Karush-Kuhn Tucker (KKT) conditions and semidefinite programming (SDP) relaxations provide effective tools for solving QCQPs(Elloumi & Lambert,2019), offering both exact and approximate solutions depending on the problem’s structure. These methods allow for efficient handling of the challenges posed by quadratic constraints and non-linearities.

Looking forward, there are several potential areas for improvement in QCQP algorithms. One direction is the development of more efficient relaxation techniques for solving non-convex QCQPs, especially in large-scale problems where computational efficiency becomes critical. Additionally, there is ongoing research into hybrid methods that combine the strengths of different optimization techniques, such as SDP and machine learning, to improve the robustness and speed of solving QCQPs in dynamic environments. As optimization problems become increasingly complex and data-rich, advancements in QCQP algorithms will continue to play a crucial role in making informed, optimal decisions in diverse applications.

Reference

[1] Agarwal, D., Singh, P., & El Sayed, M. A. (2023). The Karush–Kuhn–Tucker (KKT) optimality conditions for fuzzy-valued fractional optimization problems. Mathematics and Computers in Simulation, 205, 861-877, DOI:10.1016/j.matcom.2022.10.024

[2] Bao, X., Sahinidis, N. V., & Tawarmalani, M. (2011). Semidefinite relaxations for quadratically constrained quadratic programming: A review and comparisons (PDF). Mathematical programming, 129, 129-157.

[3] Bose, S., Gayme, D. F., Chandy, K. M., & Low, S. H. (2015). Quadratically constrained quadratic programs on acyclic graphs with application to power flow. IEEE Transactions on Control of Network Systems, 2(3), 278-287,DOI: 10.1109/TCNS.2015.2401172

[4] Elloumi, S., & Lambert, A. (2019). Global solution of non-convex quadratically constrained quadratic programs. Optimization methods and software, 34(1), 98-114,doi: https://doi.org/10.1080/10556788.2017.1350675

[5] Freund, R. M. (2004). Introduction to semidefinite programming (SDP) (PDF). Massachusetts Institute of Technology, 8-11.

[6] Ghojogh, B., Ghodsi, A., Karray, F., & Crowley, M. (2021). KKT conditions, first-order and second-order optimization, and distributed optimization: tutorial and survey. arXiv preprint arXiv:2110.01858.

[7] McCarl, B. A., Moskowitz, H., & Furtan, H. (1977). Quadratic programming applications. Omega, 5(1), 43-55.

[8] Zenios, S. A. (Ed.). (1993). Financial optimization. Cambridge university press,doi:https://doi.org/10.1017/CBO9780511522130

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Category: NonLinear Programming (NLP) - Quadratic programming

Quadratic constrained quadratic programming

Contents

Introduction

Algorithm Discussion