Applied Lagrange Duality for Constrained Optimization

Applied Lagrange Duality for Constrained Optimization February 10, 2004 1 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 variables” 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” variables 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” variables. 2 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 exploited in computing solutions to both the primal and the dual problem. • Other uses, too . . . . 3 Review of Convexity 3.1 Local and Global Optima of a Function The ball centered at x̄ with radius
is the set: B(¯ x,
) := {x|x − x ¯ ≤
}. Consider the following optimization problem over the set F: P : minx or maxx f (x) s.t. x 3 ∈ F We have the following definitions of local/global, strict/non-strict minima/maxima. Definition 3.1 x ∈ F is a local minimum of P if there exists
> 0 such that f (x) ≤ f (y) for all y ∈ B(x,
) ∩ F. Definition 3.2 x ∈ F is a global minimum of P if f (x) ≤ f (y) for all y ∈ F. Definition 3.3 x ∈ F is a strict local minimum of P if there exists
> 0 such that f (x) < f (y) for all y ∈ B(x,
) ∩ F, y =  x. Definition 3.4 x ∈ F is a strict global minimum of P if f (x) < f (y) for all y ∈ F, y = x. Definition 3.5 x ∈ F is a local maximum of P if there exists
> 0 such that f (x) ≥ f (y) for all y ∈ B(x,
) ∩ F. Definition 3.6 x ∈ F is a global maximum of P if f (x) ≥ f (y) for all y ∈ F. Definition 3.7 x ∈ F is a strict local maximum of P if there exists
> 0 such that f (x) > f (y) for all y ∈ B(x,
) ∩ F, y = x. Definition 3.8 x ∈ F is a strict global maximum of P if f (x) > f (y) for all y ∈ F, y = x. The phenomenon of local versus global optima is illustrated in Figure 1. 3.2 Convex Sets and Functions Convex sets and convex functions play an extremely important role in the study of optimization models. We start with the definition of a convex set: Definition 3.9 A subset S ⊂ n is a convex set if x, y ∈ S ⇒ λx + (1 − λ)y ∈ S for any λ ∈ [0, 1]. 4 F Figure 1: Illustration of local versus global optima. Figure 2: Illustration of convex and non-convex sets. Figure 3: Illustration of the intersection of convex sets. 5 Figure 4: Illustration of convex and strictly convex functions. Figure 2 shows a convex set and a non-convex set. Proposition 3.1 If S, T are convex sets, then S ∩ T is a convex set. This proposition is illustrated in Figure 3. Proposition 3.2 The intersection of any collection of convex sets is a con­ vex set. We now turn our attention to convex functions, defined below. Definition 3.10 A function f (x) is a convex function if f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y) for all x and y and for all λ ∈ [0, 1]. Definition 3.11 A function f (x) is a strictly convex function if f (λx + (1 − λ)y) < λf (x) + (1 − λ)f (y) for all x and y, x = y, and for all λ ∈ (0, 1). Figure 4 illustrates convex and strictly convex functions. Now consider the following optimization problem, where the feasible region is simply described as the set F : 6 P : minimizex f (x) s.t. x ∈ F Theorem 3.1 Suppose that F is a convex set, f : F → is a convex function, and x ¯ is a global minimum of f ¯ is a local minimum of P . Then x over F. Proof: Suppose x̄ is not a global minimum, i.e., there exists y ∈ F for x). Let y(λ) = λx+(1−λ)y, which is a convex combination which f (y) < f (¯ ¯ of x̄ and y for λ ∈ [0, 1] (and therefore, y(λ) ∈ F for λ ∈ [0, 1]). Note that y(λ) → x̄ as λ → 1. From the convexity of f (·), f (y(λ)) = f (λx+(1−λ)y) ¯ ≤ λf (¯ x)+(1−λ)f (y) < λf (¯ x)+(1−λ)f (¯ x) = f (¯ x) for all λ ∈ (0, 1). Therefore, f (y(λ)) < f (¯ x) for all λ ∈ (0, 1), and so x ¯ is not a local minimum, resulting in a contradiction. q.e.d. Some examples of convex functions of one variable are: • f (x) = ax + b • f (x) = x2 + bx + c • f (x) = |x| • f (x) = − ln(x) for x > 0 • f (x) = 1 x for x > 0 • f (x) = ex 7 Figure 5: Illustration of concave and strictly concave functions. 3.3 Concave Functions and Maximization The “opposite” of a convex function is a concave function, defined below: Definition 3.12 A function f (x) is a concave function if f (λx + (1 − λ)y) ≥ λf (x) + (1 − λ)f (y) for all x and y and for all λ ∈ [0, 1]. Definition 3.13 A function f (x) is a strictly concave function if f (λx + (1 − λ)y) > λf (x) + (1 − λ)f (y) for all x and y, x = y, and for all λ ∈ (0, 1). Figure 5 illustrates concave and strictly concave functions. Now consider the maximization problem P : P : maximizex f (x) s.t. x ∈ F Theorem 3.2 Suppose that F is a convex set, f : F → is a concave function, and x ¯ is a local maximum of P . Then x ¯ is a global maximum of f over F. 8 3.4 Convex Programs Now consider the optimization problem P which might be either a minimization problem or a maximization problem: P : minx or maxx f (x) s.t. x ∈ F Definition 3.14 We call P a convex program if • F is a convex set, and • we are minimizing and f (x) is a convex function, or we are maximizing and f (x)is a concave function. Then from the above theorems, every local optima of the objective function of a convex program is a global optima of the objective function. The class of convex programs is the class of “well-behaved” optimization problems. 3.5 Linear Functions, Convexity, and Concavity Proposition 3.3 A linear function f (x) = aT x + b is both convex and concave. Proposition 3.4 If f (x) is both convex and concave, then f (x) is a linear function. These properties are illustrated in Figure 6. 9 Figure 6: A linear function is convex and concave. 4 A Separating Hyperplane Theorem The main tool that is used in developing duality, analyzing dual problems, etc., is the “separating hyperplane” theorem for convex sets. This theorem states that a point outside of a convex set can be separated from the set by hyperplane. The theorem is intuitively obvious, and is very easy to prove. Theorem 4.1 Strong Separating Hyperplane Theorem: Suppose that S is a convex set in n , and that we are given a point x ¯∈ / S. Then there exists a vector u =  0 and a scalar α for which the following hold: • uT x > α for all x ∈ S • uT x ¯<α This is illustrated in Figure 7. A weaker version of this theorem is as follows: Theorem 4.2 Weak Separating Hyperplane Theorem: Suppose that ¯∈ / S or x ¯ ∈ ∂S. S is a convex set in n , and that we are given a point x Then there exists a vector u = 0 and a scalar α for which the following hold: • uT x ≥ α for all x ∈ S ¯≤α • uT x 10 S –x uTx = α Figure 7: Illustration of strong separation of a point from a convex set. S –x uTx = α Figure 8: Illustration of weak separation of a point from a convex set. This is illustrated in Figure 8. Now consider the following simple primal and dual problems, that will be prototypical of the geometry of duality that we will develop in this lecture. Primal : minimumx height of x s.t. x x 11 ∈ S, ∈ L Dual : maximumπ intercept of line π with L s.t. line π lies below S This is illustrated in Figure 9. z L S π r Figure 9: Illustration intersecting line dual. 5 Definition of the Dual Problem Recall the basic constrained optimization model: 12 OP : minimumx f (x) s.t. g1 (x) · · .. . ≤ 0, = ≥ gm (x) ≤ 0, x ∈ P, In this model, we have f (x) : n → and gi (x) : n → , i = 1, . . . , m. Of course, we can always convert the constraints to be “≤”, and so for now we will presume that our problem is of the form: OP : z ∗ = minimumx f (x) s.t. g1 (x) ≤ 0, .. . gm (x) x ∈ P, Here P could be any set, such as: • P = n • P = {x | x ≥ 0} 13 ≤ 0, • P = {x | gi (x) ≤ 0, i = m + 1, . . . , m + k}
n • P = x | x ∈ Z+  • P = {x | Ax ≤ b} Let z ∗ denote the optimal objective value of OP. For a nonnegative vector u, form the Lagrangian function: L(x, u) := f (x) + uT g(x) = f (x) + m  ui gi (x) i=1 by taking the constraints out of the body of the model, and placing them in the objective function with costs/prices/multipliers ui for i = 1, . . . , m. We then solve the presumably easier problem: L∗ (u) := minimumx f (x) + uT g(x) x∈P s.t. The function L∗ (u) is called the dual function. We presume that computing L∗ (u) is an easy task. The dual problem is then defined to be: D : v ∗ = maximumu L∗ (u) s.t. 14 u≥0 6 Steps in the Construction of the Dual Problem We start with the primal problem: OP : z ∗ = minimumx f (x) s.t. gi (x) ≤ 0, i = 1, . . . , m, x ∈ P, Constructing the dual involves a three-step process: • Step 1. Create the Lagrangian L(x, u) := f (x) + uT g(x) . • Step 2. Create the dual function: L∗ (u) := minimumx f (x) + uT g(x) s.t. x∈P • Step 3. Create the dual problem: D : v ∗ = maximumu L∗ (u) s.t. 15 u≥0 7 Examples of Dual Constructions of Optimization Problems 7.1 The Dual of a Linear Problem Consider the linear optimization problem: LP : minimumx cT x Ax ≥ b s.t. What is the dual of this problem? 7.2 The Dual of a Binary Integer Problem Consider the binary integer problem: IP : minimumx cT x s.t. Ax ≥ b xj ∈ {0, 1}, j = 1, . . . , n . What is the dual of this problem? 7.3 The Dual of a Log-Barrier Problem Consider the following logarithmic barrier problem: BP : minimumx1 ,x2 ,x3 5x1 + 7x2 − 4x3 − 3  j=1 ln(xj ) x1 + 3x2 + 12x3 = 37 s.t. x1 > 0, x2 > 0, x3 > 0 . What is the dual of this problem? 16 7.4 The Dual of a Quadratic Problem Consider the quadratic optimization problem: QP : minimumx 1 T 2 x Qx + cT x Ax ≥ b s.t. where Q is SPSD (symmetric and positive semi-definite). What is the dual of this problem? 7.5 Remarks on Problems with Different Formats of Constraints Suppose that our problem has some inequality and equality constraints. Then just as in the case of linear optimization, we assign nonnegative dual variables ui to constraints of the form gi (x) ≤ 0, unrestricted dual variables ui to equality constraints gi (x) = 0, and non-positive dual variables ui to constraints of the form gi (x) ≥ 0. For example, suppose our problem is: OP : minimumx f (x) s.t. gi (x) ≤ 0, i ∈ L gi (x) ≥ 0, i ∈ G gi (x) = 0, i ∈ E x ∈ P, Then we form the Lagrangian: 17 L(x, u) := f (x) + uT g(x) = f (x) +  ui gi (x) + i∈L  ui gi (x) + i∈G  ui gi (x) i∈E and construct the dual function L∗ (u): L∗ (u) := minimumx f (x) + uT g(x) x∈P s.t. The dual problem is then defined to be: D : maximumu L∗ (u) s.t. ui ≥ 0, i ∈ L ui ≤ 0, i ∈ G > ui <, i ∈ E For simplicity, we presume that the constraints of OP are of the form gi (x) ≤ 0. This is only for ease of notation, as the results we develop pertain to the general case just described. 8 The Dual is a Concave Maximization Problem We start with the primal problem: 18 OP : minimumx f (x) s.t. gi (x) ≤ 0, i = 1, . . . , m x ∈ P, We create the Lagrangian: L(x, u) := f (x) + uT g(x) and the dual function: L∗ (u) := minimumx f (x) + uT g(x) x∈P s.t. The dual problem then is: D : maximumu L∗ (u) s.t. u≥0 Theorem 8.1 The dual function L∗ (u) is a concave function. Proof: Let u1 ≥ 0 and u2 ≥ 0 be two values of the dual variables, and let 19 u = λu1 + (1 − λ)u2 , where λ ∈ [0, 1]. Then L∗ (u) = minx∈P f (x) + uT g(x)       = minx∈P λ f (x) + uT1 g(x) + (1 − λ) f (x) + uT2 g(x)   ≥ λ minx∈P f (x) + uT1 g(x) + (1 − λ) minx∈P (f (x) + uT2 g(x) = λL∗ (u1 ) + (1 − λ)L∗ (u2 ) . Therefore we see that L∗ (u) is a concave function. q.e.d. 9 Weak Duality Let z ∗ and v ∗ be the optimal values of the primal and the dual problems: OP : z ∗ = minimumx f (x) s.t. gi (x) ≤ 0, i = 1, . . . , m x ∈ P, D: v ∗ = maximumu L∗ (u) s.t. u≥0 Theorem 9.1 Weak Duality Theorem: If x ¯ is feasible for OP and u ¯ is feasible for D, then u) f (¯ x) ≥ L∗ (¯ 20 In particular, z∗ ≥ v∗ . Proof: If x ¯ is feasible for OP and u ¯ is feasible for D, then x) ≥ min f (x) + u ¯T g(x) = L∗ (¯ u) . f (¯ x) ≥ f (¯ x) + u ¯T g(¯ x∈P Therefore z ∗ ≥ v ∗ . q.e.d. 10 The Column Geometry of the Primal and Dual Problems For this section only, we will assume that our constraints are all equality constraints, and so our problems are: OP : z ∗ = minimumx f (x) s.t. gi (x) = 0, i = 1, . . . , m x ∈ P, D: v ∗ = maximumu L∗ (u) s.t. u<0 > where L∗ (u) := minimumx f (x) + uT g(x) x∈P s.t. 21 Let us consider the primal problem from a “resource-cost” point of view. For each x ∈ P , we have an array of resources and costs associated with x, namely:    r z r1 r2 .. .   g1 (x) g2 (x) .. .           g(x)      = . = =     f (x)  rm   g (x)  m z f (x) We can think of each of these arrays as an array of resources and cost in If we plot all of these arrays over all values of x ∈ P , we will obtain a region in m+1 , which we will call the region S: m+1 .  m+1 S := (r, z) ∈ | (r, z) = (g(x), f (x)) for some x ∈ P  . This region is illustrated in Figure 10. L = {( r, z ) | r = 0 } z S H r1 r=0 r2 Figure 10: The column geometry of the primal and dual problem. 22 Where is the feasible region of the primal in this problem? Well, we are in the wrong space to observe this. But notice that the figure contains the origin and a vertical line L going up from the origin. The points on this vertical line have resource utilization of 0, and their costs are their heights on the vertical line. Therefore, every point in the intersection of S and L corresponds to a feasible solution, and the feasible solution with the lowest cost corresponds to the lowest point on the intersection of S and L. We re-state this for emphasis: Every point in the intersection of S and L corresponds to a feasible solu­ tion of OP, and the feasible solution with the lowest cost corresponds to the lowest point in the intersection of S and L. This lowest value is exactly the value of the primal problem, namely z ∗ . In Figure 10, notice that there is also drawn a hyperplane H, and that this hyperplane H lies below the region S. Suppose that we try to find that hyperplane H lying below S whose intersection with L is as large as possible. Let us formulate this problem mathematically. We will let  H = Hu,α = (r, z) ∈ The line L is:  L = (r, z) ∈ m+1 m+1 | z + uT r = α  .  |r=0 . The intersection of Hu,α with L is: H ∩L and it occurs at the point (r, z) = (0, α) . Then the problem of determining the hyperplane Hu,α lying below S whose intersection with L is the highest is: 23 maximumu,α α s.t. Hu,α lies below S = maximumu,α s.t. = maximumu,α s.t. = maximumu,α = α z + uT r ≥ α, for all (r, z) ∈ S ≥ α, for all x ∈ P α f (x) + uT g(x) α s.t. L∗ (u) maximumu L∗ (u) ≥ α, and this last expression is exactly the dual problem (for the special case of this section). Therefore: The dual problem corresponds to finding that hyperplane Hu,α lying below S whose intersection with L is the highest. This highest value is exactly the value of the dual problem, namely v ∗ . 24 11 Strong Duality When the set S is a convex set, then generally speaking, we would expect z ∗ = v ∗ , that is, the primal and the dual have identical optimal objective function values. When the set S is not convex, we would not necessarily expect this to happen, see the illustrations in Figures 11 and 12. z L S H r r=0 Figure 11: The column geometry of the dual problem when S is convex. z L z* v* S H r Figure 12: The column geometry of the dual problem when S is not convex. As it turns out, convexity is the key to strong duality. When the problem OP is a convex program, then strong duality will hold in general, as the following theorem states: 25 Theorem 11.1 Strong Duality Theorem: Suppose that OP is a convex optimization problem, that is, all constraints of OP are of the form gi (x) ≤ 0, i = 1, . . . , m, f (x) as well as gi (x) are convex functions, i = 1, . . . , m, and P is a convex set. Then under very mild additional conditions, z ∗ = v ∗ . 12 Duality Strategies 12.1 Dualizing “Bad” Constraints Suppose we wish to solve: OP : minimumx cT x s.t. ≤ Ax b Nx ≤ g . Suppose that optimization over the constraints “N x ≤ g” is easy, but that the addition of the constraints “Ax ≤ b” makes the problem much more difficult. This can happen, for example, when the constraints “N x ≤ g” are network constraints, and when “Ax ≤ b” are non-network constraints in the model. Let P = {x | N x ≤ g} and re-write OP as: OP : minimumx cT x s.t. Ax x The Lagrangian is: 26 ≤ b ∈ P . L(x, u) = cT x + uT (Ax − b) = −uT b + (cT + uT A)x , and the dual function is: L∗ (u) := minimumx −uT b + (cT + uT A)x s.t. ∈ P . x Notice that L∗ (u) is easy to evaluate for any value of u, and so we can attempt to solve OP by designing an algorithm to solve the dual problem: D : maximumu s.t. 12.2 L∗ (u) u≥0. Dualizing A Large Problem into Many Small Problems Suppose we wish to solve: OP : minimumx1 ,x2 s.t. (c1 )T x1 +(c2 )T x2 B 1 x1 +B 2 x2 A1 x1 ≤ d ≤ b1 A2 x2 ≤ b2 Notice here that if it were not for the constraints “B 1 x1 + B 2 x2 ≤ d”, that we would be able to separate the problem into two separate problems. Let us dualize on these constraints. Let:  P = (x1 , x2 ) | A1 x1 ≤ b1 , A2 x2 ≤ b2 27  and re-write OP as: OP : minimumx1 ,x2 (c1 )T x1 +(c2 )T x2 B 1 x1 s.t. +B 2 x2 ≤ (x1 , x2 ) ∈ P d The Lagrangian is: L(x, u) = (c1 )T x1 + (c2 )T x2 + uT (B 1 x1 + B 2 x2 − d) = −uT d + ((c1 )T + uT B 1 )x1 + ((c2 )T + uT B 2 )x2 , and the dual function is: L∗ (u) = minimumx1 ,x2 −uT d + ((c1 )T + uT B 1 )x1 + ((c2 )T + uT B 2 )x2 (x1 , x2 ) s.t. which can be re-written as: L∗ (u) = −uT d + minimumA1 x1 ≤b1 ((c1 )T + uT B 1 )x1 + minimumA2 x2 ≤b2 ((c2 )T + uT B 2 )x2 Notice once again that L∗ (u) is easy to evaluate for any value of u, and so we can attempt to solve OP by designing an algorithm to solve the dual problem: D : maximumu L∗ (u) s.t. 28 u≥0 ∈ P 13 Illustration of Lagrange Duality in Discrete Optimization In order to suggest the computational power of duality theory and to illustrate duality constructs in discrete optimization, let us consider the simple constrained shortest path problem portrayed in Figure 13. 2 1 4 1 1 1 1 3 1 1 6 3 3 2 5 Figure 13: A constrained shortest path problem. The objective in this problem is to find the shortest (least cost) path from node 1 to node 6 subject to the constraint that the chosen path uses exactly four arcs. In an applied context the network might contain millions of nodes and arcs and billions of paths. For example, in one application setting that can be solved using the ideas considered in this discussion, each arc would have both a travel cost and travel time (which for our example is 1 for each arc). The objective is to find the least cost path between a given pair of nodes from all those paths whose travel time does not exceed a prescribed limit (which in our example is 4 and must be met exactly). Table 1 shows that only three paths, namely 1-2-3-4-6, 1-2-3-5-6, and 1-3-5-4-6 between nodes 1 and 6 contain exactly four arcs. Since path 1-35-4-6 has the least cost of these three paths (at a cost of 5), this path is the optimal solution of the constrained shortest path problem. Now suppose that we have available an efficient algorithm for solving 29 Path 1-2-4-6 1-3-4-6 1-3-5-6 1-2-3-4-6 1-2-3-5-6 1-3-5-4-6 1-2-3-5-4-6 Number of Arcs 3 3 3 4 4 4 5 Cost 3 5 6 6 7 5 6 Table 1: Paths from node 1 to node 6. (unconstrained) shortest path problems (indeed, such algorithms are readily available in practice), and that we wish to use this algorithm to solve the given problem. Let us first formulate the constrained shortest path problem as an integer program: CSP P : z ∗ = minxij s.t. x12 +x13 +x23 +x24 +3x34 +2x35 +x54 +x46 +3x56 x12 +x13 +x23 +x24 +x34 +x35 +x54 +x46 +x56 x12 +x13 = 1 −x23 −x24 x12 x13 = 4 = 0 −x34 +x23 −x35 +x54 −x46 +x34 x24 = 0 x35 −x54 x46 xij ∈ {0, 1} for all xij . 30 = 0 −x56 = 0 +x56 = 1 In this formulation, xij is a binary variable indicating whether or not arc i − j is chosen as part of the optimal path. The first constraint in this formulation specifies that the optimal path must contain exactly four arcs. The remaining constraints are the familiar constraints describing a shortest path problem. They indicate that one unit of flow is available at node 1 and must be sent to node 6; nodes 2, 3, 4, and 5 act as intermediate nodes that neither generate nor consume flow. Since the decision variables xij must be integer, any feasible solution to these constraints traces out a path from node 1 to node 6. The objective function determines the cost of any such path. Notice that if we eliminate the first constraint from this problem, then the problem becomes an unconstrained shortest path problem that can be solved by our available algorithm. Suppose, for example, that we dualize the first (the complicating) constraint by moving it to the objective function multiplied by a Lagrange multiplier u. The objective function then becomes: minimumxij   cij xij + u 4 − i,j   xij  i,j and collecting the terms, the modified problem becomes: 31 L∗ (u) = minxij (1 − u)x12 + (1 − u)x13 + (1 − u)x23 +(1 − u)x24 + (3 − u)x34 + (2 − u)x35 +(1 − u)x54 + (1 − u)x46 + (3 − u)x56 + 4u s.t. x12 +x13 = 1 −x23 −x24 x12 x13 = 0 −x34 −x35 +x23 +x54 −x46 +x34 x24 = 0 x35 −x54 −x56 = 0 x46 xij ∈ {0, 1} for all xij = 0 +x56 = 1 . Let L∗ (u) denote the optimal objective function value of this problem, for any given multiplier u. Notice that for a fixed value of u, 4u is a constant and the problem becomes a shortest path problem with modified arc costs given by � cij = cij − u . Note that for a fixed value of u, this problem can be solved by our available algorithm for the shortest path problem. Also notice that for any fixed value of u, L∗ (u) is a lower bound on the optimal objective value z ∗ to the constrained shortest path problem defined by the original formulation. To obtain the best lower bounds, we need to solve the Lagrangian dual problem: 32 D : v ∗ = maximumu L∗ (u) < s.t. u>0 To relate this problem to the earlier material on duality, let the decision set P be given by the set of feasible paths, namely the solutions of: x12 +x13 = 1 −x23 −x24 x12 x13 = 0 −x34 −x35 +x23 +x54 −x46 +x34 x24 = 0 x35 −x54 −x56 = 0 x46 xij ∈ {0, 1} for all xij = 0 +x56 = 1 . Of course, this set will be a finite (discrete) set. That is, P is the set of paths connecting node 1 to node 6. Suppose that we plot the cost z and the number of arcs r used in each of these paths, see Figure 14. We obtain the seven dots in this figure corresponding to the seven paths listed in Table 1. Now, because our original problem was formulated as a minimization problem, the geometric dual is obtained by finding the line that lies on or below these dots and that has the largest intercept with the feasibility line. Note from Figure 14 that the optimal value of the dual is v ∗ = 4.5, whereas the optimal value of the original primal problem is z ∗ = 5. In this example, because the cost-resource set is not convex (it is simply the seven dots), we encounter a duality gap. In general, integer optimization 33 conv(S) 7 z (cost) 6 z* = 5 v* = 4.5 5 4 3 2 1 1 2 3 4 5 6 7 r (number of arcs) Figure 14: Resources and costs. models will usually have such duality gaps, but they are typically not very large in practice. Nevertheless, the dual bounding information provided by v ∗ ≤ z ∗ can be used in branch and bound procedures to devise efficient algorithms for solving integer optimization models. It might be useful to re-cap. The original constrained shortest path problem might be very difficult to solve directly (at least when there are hundreds of thousands of nodes and arcs). Duality theory permits us to relax (eliminate) the complicating constraint and to solve a much easier � Lagrangian shortest path problem with modified arc costs cij = cij − u. Solving for any u generates a lower bound L∗ (u) on the value z ∗ to the original problem. Finding the best lower bound maxu L∗ (u) requires that we develop an efficient procedure for solving for the optimal u. Duality theory permits us to replace a single and difficult problem (in this instance, a constrained shortest-path problem) with a sequence of much easier problems (in this instance, a number of unconstrained shortest path problems), one for each value of u encountered in our search procedure. Practice on a wide variety of problem applications has shown that the dual problem and subsequent branch and bound procedures can be extremely effective in many 34 applications. To conclude this discussion, note from Figure 14 that the optimal value for u in the dual problem is u = 1.5. Also note that if we subtract u = 1.5 from the cost of every arc in the network, then the shortest path (with respect to the modified costs) has cost −1.5. Both paths 1-2-4-6 and 1-2-35-4-6 are shortest paths for this modified problem. Adding 4u = 4 × 1.5 = 6 to the modified shortest path cost of −1.5 gives a value of 6 − 1.5 = 4.5 which equals v ∗ . Note that if we permitted the flow in the original problem to be split among multiple paths (i.e., formulate the problem as a linear optimization problem instead of as an integer optimization problem), then the optimal solution would send one half of the flow on both paths 1-2-4-6 and 1-2-3-5-4-6 and incur a cost of: 1 1 × 3 + × 6 = 4.5 . 2 2 In general, the value of the dual problem is always identical to the value of the original (primal) problem if we permit convex combinations of primal solutions. This equivalence between convexification and dualization is easy to see geometrically. 35