Glossary

13.1 Formulation of quadratic programs

3 min read•july 30, 2024

Quadratic programming builds on linear programming by introducing quadratic terms in the . This allows for modeling more complex relationships between variables, like diminishing returns or trade-offs between competing goals.

The standard form includes a with . Key components are the Q matrix for quadratic coefficients, c vector for linear terms, and Ax ≤ b for constraints. This versatile framework applies to many real-world optimization problems.

Quadratic Programming Standard Form

Mathematical Representation

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  • Minimize or maximize quadratic objective function f(x)=12xTQx+cTxf(x) = \frac{1}{2}x^T Q x + c^T x subject to linear constraints
  • QQ represents symmetric matrix of quadratic coefficients
  • cc denotes vector of linear coefficients
  • xx signifies vector of
  • Linear constraints expressed as Ax≤bAx \leq b
    • AA represents matrix of constraint coefficients
    • bb denotes vector of constraint constants
  • Non-negativity constraints x≥0x \geq 0 ensure of solution

Characteristics and Special Cases

  • Feasible region formed by intersection of all constraints creates convex set
  • Positive semi-definite QQ matrices ensure of objective function
  • Convexity guarantees global optimality of local minimum ()
  • Indefinite QQ matrices may lead to multiple local optima (facility location problems)

Objective Function and Constraints

Objective Function Components

  • Quadratic expression minimized or maximized contains linear and quadratic terms
  • QQ matrix represents quadratic interactions between variables (production costs)
  • cc vector represents linear coefficients of decision variables (profit margins)
  • Objective function models complex relationships (diminishing returns in marketing)
  • Captures trade-offs between competing objectives (risk vs. return in finance)

Constraint Types and Formulation

  • Linear inequalities or equalities restrict feasible region of solution
  • Equality constraints expressed as Ax=bAx = b ()
  • Inequality constraints expressed as Ax≤bAx \leq b or Ax≥bAx \geq b (production limits)
  • Bound constraints limit individual variables (inventory levels)
  • Separating bound constraints from general linear constraints improves computational efficiency

Real-World Problems in Quadratic Form

Problem Formulation Process

  • Identify decision variables representing quantities to be optimized (production quantities)
  • Formulate objective function expressing goal in terms of decision variables
  • Include both linear and quadratic terms (profit maximization with diminishing returns)
  • Determine constraints by identifying limitations or requirements (resource availability)
  • Express constraints as linear equalities or inequalities (machine capacity limits)
  • Convert all expressions to matrix and vector notation aligning with standard form
  • Verify formulation accurately represents original problem (portfolio optimization)

Common Applications

  • Portfolio optimization balancing risk and return
  • Production planning with economies of scale
  • Facility location problems considering distance and demand
  • Traffic flow optimization in transportation networks
  • Resource allocation in project management
  • Machine learning algorithms (support vector machines)

Linear vs Quadratic Programming

Objective Function Differences

  • Linear programming involves linear objective function (maximize profit)
  • Quadratic programming includes both linear and quadratic terms (minimize risk)
  • Quadratic terms model more complex relationships between variables (economies of scale)
  • Quadratic programming captures diminishing returns or increasing costs (marketing effectiveness)

Solution Characteristics

  • Linear programming optimal solutions always occur at vertices of feasible region
  • Quadratic programming optimal solutions may lie in interior of feasible region
  • Interior solutions in quadratic programming reflect trade-offs between competing objectives
  • Quadratic programming solutions often more realistic for real-world scenarios (portfolio allocation)

Computational Complexity

  • Linear programming solved efficiently using simplex method or interior point methods
  • Quadratic programming requires more complex algorithms (active set methods)
  • Convexity of quadratic programming problems crucial for global optimality
  • Non-convex quadratic programs may have multiple local optima, requiring global optimization techniques

Key Terms to Review (16)

Concave: A function is considered concave if, for any two points on its curve, the line segment connecting these points lies below or on the graph of the function. This property indicates that the function curves downwards, and it is critical in optimization as it ensures that any local maximum is also a global maximum, facilitating easier solutions in quadratic programs.
Constraint set: The constraint set is a collection of all possible values that satisfy the given constraints in an optimization problem. This set defines the feasible region within which solutions must lie, ensuring that any optimal solution adheres to the specified limitations, whether they are in the form of equations or inequalities. Understanding the constraint set is crucial because it directly impacts the feasibility and optimality of potential solutions in quadratic programming.
Convexity: Convexity refers to a property of a set or a function where, for any two points within the set or on the function, the line segment connecting these points lies entirely within the set or above the function. This characteristic is crucial in optimization as it simplifies the analysis of feasible regions, objective functions, and constraints, leading to more efficient solution methods.
Cvx: 'Cvx' refers to convexity in the context of optimization problems. A set or function is convex if, for any two points within it, the line segment connecting these points lies entirely within the set or above the function. This property is crucial in optimization because convex problems are easier to solve, guaranteeing that any local minimum is also a global minimum, making them highly desirable for formulation in quadratic programming and semidefinite programming.
Decision Variables: Decision variables are the unknown values in an optimization problem that decision-makers seek to determine. They represent the choices available in a model and are essential for formulating the problem, as they directly impact the objective function and constraints that govern the system being optimized.
Dual Simplex Method: The dual simplex method is an optimization technique used to solve linear programming problems by iterating on the dual feasible region while maintaining primal feasibility. It is particularly useful for problems where the primal solution becomes infeasible due to changes in constraints but where dual feasibility can still be achieved, such as in sensitivity analysis and problems involving resource allocation. This method connects deeply with the simplex algorithm and revised simplex method, highlighting special cases like degeneracy, unboundedness, and infeasibility, and extends into quadratic programming and semidefinite programming.
Feasibility: Feasibility refers to the condition of a solution or set of solutions that satisfies all constraints of an optimization problem. It is essential to determine whether a proposed solution can be realized under given restrictions, such as resource limitations and requirements for decision variables, thereby connecting the solution space to valid and practical outcomes.
Karush-Kuhn-Tucker Conditions: The Karush-Kuhn-Tucker (KKT) conditions are a set of mathematical equations and inequalities that provide necessary and sufficient conditions for a solution to be optimal in constrained optimization problems. They extend the method of Lagrange multipliers to handle both equality and inequality constraints, making them essential in various optimization scenarios.
Linear Constraints: Linear constraints are mathematical expressions that define the conditions or limitations placed on the decision variables in optimization problems, typically represented as linear inequalities or equalities. They help establish a feasible region where potential solutions exist and influence the optimization of an objective function. By specifying the limits within which solutions can be found, linear constraints play a critical role in determining optimal outcomes and ensuring that solutions meet practical requirements.
Matlab: Matlab is a high-level programming language and interactive environment used primarily for numerical computing, data analysis, and visualization. It provides tools for solving mathematical problems, especially in optimization, making it a popular choice in various fields such as engineering, finance, and scientific research.
Objective Function: An objective function is a mathematical expression that defines the goal of an optimization problem, representing either a maximization or minimization task. It is typically formulated as a function of decision variables, which are the unknowns that need to be determined in order to achieve the best outcome based on given constraints.
Optimal Solution: An optimal solution refers to the best possible outcome or result for a given optimization problem, maximizing or minimizing an objective function while satisfying all constraints. Finding this solution is central to various mathematical modeling techniques, as it determines the most efficient or effective way to achieve goals under specified conditions.
Portfolio optimization: Portfolio optimization is the process of selecting the best mix of assets to maximize returns while minimizing risk, typically guided by financial theories and mathematical models. This concept involves balancing expected returns against potential risks, ensuring that investments align with an investor's risk tolerance and financial goals.
Positive Definite: A matrix is said to be positive definite if it is symmetric and all its eigenvalues are positive. This property indicates that the associated quadratic form produces only positive values for all non-zero vectors. Positive definite matrices play a crucial role in optimization, as they ensure that a quadratic function has a unique minimum point, providing the necessary conditions for optimization problems to be well-posed and solvable.
Quadratic objective function: A quadratic objective function is a specific type of mathematical expression used in optimization problems, characterized by a polynomial of degree two. This function often takes the form $$f(x) = ax^2 + bx + c$$, where 'a', 'b', and 'c' are constants and 'x' is the variable being optimized. Quadratic objective functions are central to quadratic programming, which involves finding the maximum or minimum values of these functions subject to certain constraints.
Resource Allocation: Resource allocation refers to the process of distributing available resources among various projects, departments, or initiatives to optimize efficiency and effectiveness. This concept is crucial in decision-making and optimization, as it involves determining the best way to utilize limited resources, such as time, money, or manpower, to achieve specific goals.
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