13.2 Optimality conditions for quadratic programming

Quadratic programming optimizes quadratic functions with linear constraints. KKT conditions provide necessary and sufficient criteria for optimality in these problems. They consist of stationarity, complementary slackness, primal feasibility, and dual feasibility.

Understanding KKT conditions is crucial for solving quadratic programs. They help identify optimal solutions, analyze sensitivity, and provide insights into problem structure. Mastering these conditions equips you with powerful tools for tackling optimization challenges in various fields.

KKT Conditions for Quadratic Programming

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• KKT conditions serve as necessary and sufficient conditions for optimality in quadratic programming problems with convex objective functions and linear constraints
• Four main components comprise KKT conditions stationarity, complementary slackness, primal feasibility, and dual feasibility
• Stationarity condition requires gradient of Lagrangian function with respect to decision variables equals zero at optimal point
• Complementary slackness condition dictates for each constraint, either constraint is active or corresponding Lagrange multiplier is zero
• Primal feasibility condition ensures all constraints of original problem are satisfied
• Dual feasibility condition requires all Lagrange multipliers associated with inequality constraints are non-negative
• Express KKT conditions mathematically as system of equations and inequalities involving decision variables, Lagrange multipliers, and constraint functions
• Mathematically, stationarity condition $\nabla f(x^*) + \sum_{i=1}^m \lambda_i \nabla g_i(x^*) + \sum_{j=1}^p \mu_j \nabla h_j(x^*) = 0$
• Complementary slackness condition $\lambda_i g_i(x^*) = 0$ for all $i = 1, ..., m$
• Primal feasibility conditions $g_i(x^*) \leq 0$ for all $i = 1, ..., m$ and $h_j(x^*) = 0$ for all $j = 1, ..., p$
• Dual feasibility condition $\lambda_i \geq 0$ for all $i = 1, ..., m$

Application to Convex Quadratic Programming

• KKT conditions particularly useful for convex quadratic programming problems
• Convex quadratic programming problem formulation $\min_{x} \frac{1}{2}x^T Q x + c^T x$ subject to $Ax \leq b$ and $Ex = d$
• Apply KKT conditions to convex quadratic programming problem
  • Stationarity $Qx^* + c + A^T \lambda + E^T \mu = 0$
  • Complementary slackness $\lambda_i (A_i x^* - b_i) = 0$ for all $i$
  • Primal feasibility $Ax^* \leq b$ and $Ex^* = d$
  • Dual feasibility $\lambda \geq 0$
• KKT conditions provide systematic approach to identify local optima in quadratic programming
• Local optima identified through KKT conditions are also global optima due to problem's convexity

Interpretation of KKT Conditions

Economic and Geometric Interpretations

• Stationarity condition represents balance between objective function's gradient and weighted sum of constraint gradients at optimal point
• Interpret stationarity condition as equilibrium state where marginal changes in decision variables do not improve objective value
• Complementary slackness indicates which constraints are active (binding) at optimal solution and which Lagrange multipliers are non-zero
• Interpret complementary slackness geometrically constraint is either satisfied with equality (active) or its Lagrange multiplier is zero (inactive)
• Primal feasibility ensures optimal solution satisfies all original problem constraints, maintaining solution's validity
• Interpret primal feasibility as guarantee that optimal solution lies within feasible region defined by constraints
• Dual feasibility guarantees Lagrange multipliers associated with inequality constraints are non-negative, reflecting their role as shadow prices
• Interpret non-negative Lagrange multipliers as marginal cost or benefit of relaxing corresponding constraint

Insights from KKT Conditions

• KKT conditions collectively describe characteristics of optimal solution, considering both objective function and constraints simultaneously
• Provide insights into sensitivity of optimal objective value to changes in constraint parameters
• Lagrange multipliers in quadratic programming interpreted as shadow prices, indicating rate of change in objective value per unit change in constraint right-hand side
• KKT conditions help identify binding constraints at optimal solution, crucial for understanding problem's structure and sensitivity analysis
• Offer framework for analyzing trade-offs between objective improvement and constraint satisfaction
• Enable identification of redundant constraints those with zero Lagrange multipliers at optimal solution
• Facilitate understanding of how changes in problem parameters (coefficients, right-hand sides) affect optimal solution and objective value

Applying KKT Conditions for Optimality

Verification Process

• Verify stationarity condition by calculating gradient of Lagrangian function and ensuring it equals zero at given solution point
• Check complementary slackness condition by examining each constraint and its corresponding Lagrange multiplier, confirming their product is zero
• Assess primal feasibility by substituting solution into all constraints and confirming they are satisfied
• Evaluate dual feasibility by verifying all Lagrange multipliers associated with inequality constraints are non-negative
• Analyze second-order sufficient conditions to confirm solution is indeed minimum (for minimization problems) or maximum (for maximization problems)
• Second-order sufficient condition for minimization $z^T \nabla^2 L(x^*, \lambda^*, \mu^*) z \geq 0$ for all $z$ satisfying $\nabla g_i(x^*)^T z = 0$ for active constraints
• If all KKT conditions are satisfied, conclude given solution is optimal for quadratic programming problem

Troubleshooting and Analysis

• In cases where KKT conditions are not fully satisfied, identify which conditions are violated to understand why solution is not optimal
• Common violations include
  • Non-zero gradient of Lagrangian (stationarity violation)
  • Positive product of Lagrange multiplier and slack variable (complementary slackness violation)
  • Constraint violation (primal feasibility violation)
  • Negative Lagrange multipliers for inequality constraints (dual feasibility violation)
• Analyze violations to gain insights into how to improve solution or reformulate problem
• Use KKT conditions to guide iterative optimization algorithms (gradient descent, interior point methods) towards optimal solution
• Apply sensitivity analysis using KKT conditions to understand how small changes in problem parameters affect optimal solution

KKT Conditions vs Graphical Representation

Visual Interpretation of KKT Conditions

• Graphical representation of quadratic program in two dimensions consists of objective function contours and feasible region defined by constraints
• Optimal solution in graphical representation corresponds to point where KKT conditions are satisfied, typically occurring at vertex, edge, or interior point of feasible region
• Binding constraints in graphical representation associated with non-zero Lagrange multipliers in KKT conditions
• Gradient of objective function at optimal point perpendicular to active constraint boundaries, reflected in stationarity condition of KKT conditions
• Tangency between objective function contour and feasible region boundary at optimal point illustrates balance described by KKT conditions
• For problems with inequality constraints, graphical representation helps visualize complementary slackness condition, showing which constraints are active at optimal solution
• Convexity of objective function and linear constraints in graphical representation ensure solution satisfying KKT conditions is globally optimal

Comparative Analysis

• Graphical method limited to two-dimensional problems, while KKT conditions applicable to higher-dimensional quadratic programs
• KKT conditions provide precise mathematical characterization of optimal solution, whereas graphical method offers intuitive visual understanding
• Graphical representation useful for understanding problem structure and visualizing feasible region, complementing algebraic nature of KKT conditions
• KKT conditions enable systematic sensitivity analysis, while graphical method provides qualitative insights into solution behavior
• Combine graphical representation with KKT conditions for comprehensive understanding of quadratic programming problems
  • Use graphical method to gain intuition and visualize problem
  • Apply KKT conditions to rigorously verify optimality and perform detailed analysis
• In higher dimensions, visualization techniques (projections, slices) can complement KKT conditions for better problem understanding