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duality_article - Applied Lagrange Duality for Constrained...

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Applied Lagrange Duality for Constrained Optimization Robert M. Freund February 10, 2004 c ± 2004 Massachusetts Institute of Technology. 1
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1 Overview The Practical Importance of Duality Review of Convexity A Separating Hyperplane Theorem Definition of the Dual Problem Steps in the Construction of the Dual Problem Examples of Dual Constructions The Dual is a Concave Maximization Problem Weak Duality The Column Geometry of the Primal and Dual Problems Strong Duality Duality Strategies Illustration of Lagrange Duality in Discrete Optimization 2 The Practical Importance of Duality Duality arises in nonlinear (and linear) optimization models in a wide variety of settings. Some immediate examples of duality are in: Models of electrical networks. The current flows are “primal vari- ables” and the voltage differences are the “dual variables” that arise in consideration of optimization (and equilibrium) in electrical networks. Models of economic markets. In these models, the “primal” vari- ables are production levels and consumption levels, and the “dual” variables are prices of goods and services. Structural design. In these models, the tensions on the beams are “primal” variables, and the nodal displacements are the “dual” vari- ables. 2
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Nonlinear (and linear) duality is very useful. For example, dual problems and their solutions are used in connection with: Identifying near-optimal solutions. A good dual solution can be used to bound the values of primal solutions, and so can be used to actually identify when a primal solution is near-optimal. Proving optimality. Using a strong duality theorem, one can prove optimality of a primal solution by constructing a dual solution with the same objective function value. Sensitivity analysis of the primal problem. The dual variable on a constraint represents the incremental change in the optimal solution value per unit increase in the RHS of the constraint. Karush-Kuhn-Tucker (KKT) conditions. The optimal solution to the dual problem is a vector of KKT multipliers. Convergence of improvement algorithms. The dual problem is often used in the convergence analysis of algorithms. Good Structure. Quite often, the dual problem has some good mathematical, geometric, or computational structure that can ex- ploited in computing solutions to both the primal and the dual prob- lem. Other uses, too . . . . 3 Review of Convexity 3.1 Local and Global Optima of a Function The ball centered at ¯ x with radius ± is the set: x, ± ):= { x x ¯ B x ±≤ ± } . Consider the following optimization problem over the set F : P :m i n x or max x f ( x ) s.t. x ∈F 3
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We have the following definitions of local/global, strict/non-strict min- ima/maxima.
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duality_article - Applied Lagrange Duality for Constrained...

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