Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Optimization → Nonlinear Programming ↓


Quadratic Programming


Quadratic Programming (QP) is a special type of mathematical optimization problem. It is an important topic in the field of optimization, which is widely used in many other practical applications such as finance, machine learning, and control systems. The purpose of this guide to explain the concepts, mathematics and applications of Quadratic Programming in simple terms.

What is quadratic programming?

Quadratic programming deals with the optimization (minimization or maximization) of a quadratic objective function subject to linear constraints. Let's break it down:

  • The quadratic objective function is a mathematical function where the term with the highest power is the square. In general form, it is written as:
    f(x) = 0.5 * x t qx + c t x,
    where Q is an nxn symmetric matrix, c is a vector of n real numbers, and x is a vector of variables.
  • Linear constraints are constraints that limit the values a variable can take. Their general form is:
    a ≤ b,
    where A is a matrix, x is a vector of variables, and b is a vector of constraints. Constraints can be equalities or inequalities.

Mathematical representation

The standard form of a quadratic programming problem is as follows:

Minimize: f(x) = 0.5 * x T Qx + c T x

subject to:
    axis ≤ b
    x = d
    x ≥ 0

Here:

  • x T is the transpose of the vector x .
  • Q is a positive semi-definite matrix that ensures that the problem is convex.
  • A and E are matrices representing inequality and equality constraints, respectively.
  • b and d are vectors representing constants for the inequality and equality constraints, respectively.

Understanding through simple examples

For better understanding let us consider a simple quadratic programming problem.

Example 1: Simple portfolio optimization

Imagine you have two investment options and you have to decide how much to invest in each option. The goal is to minimize the risk (variance) while achieving a certain expected return. The quadratic function represents the variance (risk) of the portfolio. can represent.

Suppose that the variance can be represented by the following quadratic equation:

Minimize: f(x) = 0.5 * (x 1 2 + 2x 1 x 2 + x 2 2 )

subject to:
    x 1 + x 2 = 1 (the sum of the allocations must be 1)
    x 1 ≥ 0
    x 2 ≥ 0

The problem asks us to find the values of x 1 and x 2 that minimize the risk, provided their sum is 1 and both are not negative.

Visualization of quadrilateral programming

Looking at quadratic functions and constraints can help understand the solution space of a problem:


    
    
    
    feasible region

In the visualization above, the light blue circle represents the quadratic nature of the function. The red lines represent the constraints. The feasible region (where solutions can exist) is the intersection of the constraints and the possible values of the objective function.

Solving quadratic programming problems

Quadratic programming problems can be solved using different methods depending on the size and complexity of the problem. Some of the common methods are as follows:

  • Analytical methods: Suitable for small problems, these methods involve calculating derivatives and setting them to zero to find the minimum point.
  • Numerical methods: For larger problems, numerical methods such as the active-set method or the interior-point method are used. These methods start by searching within the feasible region to find the optimal solution.
  • Software tools: Tools such as MATLAB, CVX, and Python's SciPy library allow for easy setup and solution of QP problems.

Applications of quadratic programming

Quadratic programming is used in various industries and fields due to its ability to efficiently handle quadratic relationships and constraints.

Finance

In finance, QP is used for portfolio optimization, where the variance (or risk) of portfolio returns is minimized, given constraints such as achieving a desired level of return.

Machine learning

In machine learning, QP is used in support vector machines for classification tasks, where it helps in finding the maximum-margin hyperplane that separates different classes.

Control system

Quadratic programming is also used in model predictive control (MPC), where it optimizes control inputs subject to system dynamics and constraints to achieve desired performance.

Deep mathematical understanding

To gain a more formal understanding of quadratic programming, let's look at some mathematical aspects:

Convexity

The matrix Q ensures that the quadratic function is convex. Convex functions have the special property where any local minimum is also a global minimum, which simplifies the problem-solving process.

For a matrix to be positive semi-definite (PSD) , all of its eigenvalues must be non-negative. This property ensures convexity.

Double problem

Quadratic programming also has a duality problem, which is similar to linear programming. Solving the duality problem often provides insight or alternative numerical solutions.

The dual of a quadratic program connects to the original problem through dual variables, and gives bounds on the objective function of the original problem.

Conclusion

Quadratic programming is a versatile and powerful method for optimization where the objective function is quadratic and subject to linear constraints. Its ability to handle large and complex problems makes it invaluable in many fields. By understanding the basic concepts, applications, and solution techniques, one can effectively tackle quadratic programming challenges in real-world scenarios.


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