Optimization under uncertainty: modeling and solution methods

Transcription

1 Optimization under uncertainty: modeling and solution methods Paolo Brandimarte Dipartimento di Scienze Matematiche Politecnico di Torino URL: Lecture 2: Refresher on Optimization Theory and Methods

2 MOTIVATION Review essential concepts in convexity: convex sets and functions; subgradients; extreme points and extreme rays of polyhedra; convex cones. Introduce some essential concepts for solution algorithms: cutting planes; penalty functions; Lagrangian multipliers and KKT conditions. Emphasize the role of duality in: decomposition methods for large-scale problems; primal-dual interior point methods.

3 REFERENCES S. Boyd, L. Vandenberghe. Convex Optimization. Cambridge University Press, The book pdf can be downloaded from boyd/cvxbook/ R.J. Vanderbei. Linear Programming: Foundations and Extensions, (3rd ed.). Springer, M.Z. Bazaraa, H.D. Sherali, G.M. Shetty. Nonlinear Programming: Theory and Algorithms, (3rd ed.). Wiley, B. Gärtner, J. Matousek. Approximation Algorithms and Semidefinite Optimization. Springer, P. Brandimarte. Numerical Methods for Finance and Economics: A Matlab-Based Introduction, (2nd ed.). Wiley, P. Brandimarte. Quantitative Methods: An Introduction for Business Management. Wiley, 2011.

4 OUTLINE 1 BITS OF CONVEX ANALYSIS 2 SOLUTION METHODS 3 DUALITY IN OPTIMIZATION 4 LINEAR PROGRAMMING 5 CONVEX OPTIMIZATION

5 Convex sets Polyhedra Convex cones Convex functions CONVEXITY Convexity is arguably the most important attribute of an optimization problem. Convexity is introduced for sets and then generalized to functions. Relevant convex sets are polyhedra and polytopes, ellipsoids, convex cones, including polar and dual cones. Convexity plays a key role in duality for linear programming, and more generally in conic programming.

6 Convex sets Polyhedra Convex cones Convex functions CONVEX SETS A set S R n is a convex if x, y S λx + (1 λ)y S, λ [0, 1]. The concept of convexity can be grasped intuitively by observing that the points of the form λx + (1 λ)y, where 0 λ 1, are simply the points on the straight line segment joining x and y. So, a set S is convex if the line joining any pair of points x, y S is contained in S.

7 Convex sets Polyhedra Convex cones Convex functions CONVEX SETS x 2 x 2 x 2 S 1 S 1 is convex, but S 2 is not. x 1 S 2 x 1 x 1 S 3 is a discrete set and it is not convex; this fact has important consequences for discrete optimization problems. It is easy to see that the intersection of convex sets is a convex set. This does not necessarily apply to the union. S 3

8 Convex sets Polyhedra Convex cones Convex functions CONVEX HULLS The convex combination of p points x 1, x 2,..., x p R n is defined as x = px µ i x i, µ 1,..., µ p 0, i=1 px µ i = 1. i=1 The requirements on the weights µ i are that the linear combination is conic (positive) and affine. Note the similarity with the expected value of a discrete random variable. Given a set S R n, the set of points that are convex combinations of points in S is the convex hull of S (denoted by [S]).

9 Convex sets Polyhedra Convex cones Convex functions CONVEX HULLS The convex hull of a generic set S is the smallest convex set containing S; it can also be regarded as the intersection of all the convex sets containing S.

10 Convex sets Polyhedra Convex cones Convex functions POLYHEDRAL SETS Polyhedral sets are essential in describing the feasible set of a Linear Programming problem. A hyperplane a T i x = b i, where b i R and a i, x R n, divides R n into two half-spaces expressed by the linear inequalities a T i x b i and a T i x b i. A polyhedron P R n is a set of points satisfying a finite collection of linear inequalities, i.e., P = x R n Ax b, where the matrix A collects row vectors a T i. A polyhedron is therefore the intersection of a finite collection of half-spaces and is convex. The feasible set of an LP problem, S = {x R n Ax = b, x 0} or S = {x R n Ax b}, is a polyhedron.

11 Convex sets Polyhedra Convex cones Convex functions POLYHEDRAL SETS Polyhedra may be bounded or not. Sometimes, a bounded polyhedron is called a polytope. There are alternative ways to describe a polyhedron.

12 Convex sets Polyhedra Convex cones Convex functions DESCRIPTION OF POLYHEDRAL SETS Definition. A point x is an extreme point of a polyhedron P if x P and it is not possible to express x as x = 1 2 x x with x, x P and x x. A polytope P is the convex hull of its extreme points x 1,..., x J (a finite set). Any point x P can be represented as: x = JX λ j x j, j=1 JX λ j = 1, λ j 0; j=1 For unbounded polyhedra, we need to define extreme rays. A vector r R n is called a ray of the polyhedron if Ar 0. If x 0 P, then P = x R n Ax b y = x 0 + λr P λ 0. It is easy to adapt the definition to the case P = {x R n Ax = b, x 0}.

13 Convex sets Polyhedra Convex cones Convex functions DESCRIPTION OF POLYHEDRAL SETS Definition. A ray r of a polyhedron P is called an extreme ray if it cannot be expressed as r = 1 2 r r 2, where r 1, r 2 are rays of P such that r 1 λr 2 for any number λ > 0. A polyhedron P can be described in terms of its extreme rays and points, in the sense that any point x P can be expressed combining extreme rays and points: x = JX KX λ j x j + µ k r k, j=1 k=1 JX λ j = 1, λ j, µ k 0. j=1 So, the feasible set of an LP problem may be characterized in terms of extreme vertices and rays. Actually, we do not need all of them to spot the optimal solution and this plays a key role in some decomposition strategies for large-scale problems (Dantzig-Wolfe).

14 Convex sets Polyhedra Convex cones Convex functions CONVEX CONES Another way to express the feasible set of an LP problem with equality constraints Ax = b is obtained by thinking A R m,n in terms of its column vectors a j R m, j = 1,..., n: A = a 1 a 2 a n... Since x 0, we are expressing the right-hand side vector b as a linear combination of columns of A nx a j x j = b, with non-negative weights x j 0. j= Thus, the feasible set is a conic combination of vectors.

15 Convex sets Polyhedra Convex cones Convex functions CONVEX CONES A set C is called a cone if for every x C and λ 0 we have λx C. Cones can be convex or not closed or not pointed or not polyhedral or not A non-polyhedral cone is the Lorentz (or ice-cream) cone: C = {(x, t) R n+1 x 2 t}. This is also called second-order cone and is a particular case of norm cone. It is important in second-order cone programs, which are relevant in robust optimization.

16 Convex sets Polyhedra Convex cones Convex functions CONVEX CONES A trivial example of cone is the positive orthant R + n. The feasible set of an LP problem is the conic hull of the columns of matrix A. We may also regard the feasible set of an LP problem as the sum of a bounded polyhedron (convex hull of extreme points) and the cone of extreme rays (recession directions). A less trivial cone is the convex cone S n + of (symmetric) positive semidefinite matrices. Indeed, if Q 1, Q 2 S n +, i.e., x T Q i x 0, for any x and i = 1, 2, it is easy to see that λ 1 Q 1 + λ 2 Q 2 S n +, for any λ 1, λ 2 0. To streamline notation, we may write Q 0 rather than Q S n +.

17 Convex sets Polyhedra Convex cones Convex functions CONVEX CONES The following concepts are also useful in stochastic, conic, and robust optimization: Given a subset K R n, its polar cone is the set K = {y R n y T x 0, x K } The polar cone is a (closed) convex cone even if K is not convex. Given a subset K R n, its dual cone is the set We see that K = K. K = {y R n y T x 0, x K }

18 Convex sets Polyhedra Convex cones Convex functions DUAL CONES: TRIVIAL EXAMPLES 1 The nonnegative orthant R n + is self-dual, i.e., (R n +) = R n +. In fact, if y 0, then we have y T x 0 for all x 0, so R n + (R n +). However, if we have a component y i < 0, then y T e i < 0, where e i is the i-th unit vector. 2 It is also easy to see that (R n ) = {0} and ({0}) = R n.

19 Convex sets Polyhedra Convex cones Convex functions DUAL CONES: POSITIVE SEMIDEFINITE CONE Dual cones can be introduced in a generalized setting by considering inner products. On the set S n we introduce the inner product using the trace of matrix product: A, B tr(ab) = nx a ij b ij The notation A B is also used; note that in general (nonsymmetric) case the inner product should be defined as tr(a T B). A notable fact is that the cone S n + is self-dual, i.e., (S n +) = S n +, since it can be shown that tr(ab) 0, A 0 B 0 i,j=1

20 Convex sets Polyhedra Convex cones Convex functions DUAL CONES: NORM CONE Given a norm on R n, we have defined the norm cone K = {(x, t) R n+1 x t}. The dual norm is defined as u = sup{u T x x 1}. For instance: The dual of the l norm is the l 1 norm: nx sup{u T x x 1} = u i = u 1. By a similar token, the dual of the l 1 norm is the l norm. The Euclidean norm l 2 is self-dual. Then, the dual cone turns out to be: K = {(u, v) R n+1 u v}. i=1

21 Convex sets Polyhedra Convex cones Convex functions CONVEX FUNCTIONS A function f : R n R, defined over a convex set S R n, is a convex function on S if, for any y and z in S, for any λ [0, 1], we have f (λy + (1 λ)z) λf (y) + (1 λ)f (z). (1) The definition can be interpreted as follows f(x) f(y) f(y)+(1 )f(z) f(z) y y+(1 )z z x

22 Convex sets Polyhedra Convex cones Convex functions CONVEX FUNCTIONS In other words, a function is convex if its epigraph, i.e., the region above the function graph, is a convex set. A further link between convex sets and convex functions is that the set S = {x R n g(x) 0} is convex if g is a convex function. If the condition (1) is satisfied with strict inequality for all y z, the function is strictly convex. A function f is concave if ( f ) is convex.

23 Convex sets Polyhedra Convex cones Convex functions CONVEX FUNCTIONS f(x) f(x) f(x) (a) x (b) x (c) x The first function is convex, whereas the second is not. Note that in the second case we have a local minimum that is not a global one. Indeed, convexity is so relevant in minimization problems because it rules out local minima (in maximization problems, concavity is relevant). The third function is a polyhedral convex function: a convex function need not be differentiable everywhere.

24 Convex sets Polyhedra Convex cones Convex functions CONVEX FUNCTIONS Convexity of functions is preserved by some operations: A linear combination of convex functions f i, f (x) = mx λ i f i (x) i=1 is a convex function if λ i 0, for every i. The pointwise maximum of convex functions f i, is convex as well. f (x) = max{f 1 (x), f 2 (x),..., f m(x)} Some function compositions also preserve convexity, such as the composition with an affine mapping: g(x) = f (Ax + b)

25 Convex sets Polyhedra Convex cones Convex functions DIFFERENTIABLE CONVEX FUNCTIONS If f is a differentiable function, it is convex (over S) if and only if f (x) f (x 0 ) + f (x 0 ) T (x x 0 ), x, x 0 S. (2) Note that the hyperplane z = f (x 0 ) + f (x 0 ) T (x x 0 ) is the usual tangent hyperplane, i.e., the first-order Taylor expansion of f at x 0. For a differentiable function, convexity implies that the first-order approximation at a certain point x 0 consistently underestimates the true value of the function at all the other points x S.

26 Convex sets Polyhedra Convex cones Convex functions SUPPORT HYPERPLANES f(x) f(x) It is easy to see why stationarity at a point x, i.e., f (x ) = 0, is a necessary and sufficient condition for optimality in the unconstrained, convex, differentiable case. The concept of a tangent hyperplane applies only to differentiable convex functions, but it can be generalized by the concept of a support hyperplane.

27 Convex sets Polyhedra Convex cones Convex functions SUBGRADIENTS AND SUBDIFFERENTIALS OF CONVEX FUNCTIONS Definition. Given a convex function f and a point x 0, the hyperplane (in R n+1 ) given by z = f (x 0 ) + γ T (x x 0 ), which meets the epigraph of f in (x 0, f (x 0 )) and lies below it, is called the support hyperplane of f at x 0. A support hyperplane at x 0 is essentially defined by a vector γ such that f (x) f (x 0 ) + γ T (x x 0 ), x S. (3) The vector γ in inequality (3) plays the same role as the gradient does in inequality (2). If f is differentiable in x 0, the support hyperplane is the usual tangent hyperplane and γ = f (x 0 ). This is why a vector γ such that inequality (3) holds is called a subgradient of f at x 0.

28 Convex sets Polyhedra Convex cones Convex functions SUBGRADIENTS AND SUBDIFFERENTIALS OF CONVEX FUNCTIONS If f is non-differentiable, the support hyperplane need not be unique and there is a set of subgradients. The set of subgradients at a point x 0 is called the subdifferential of f at x 0, and it is denoted by f (x 0 ). It can be shown that a convex function on a set S is subdifferentiable on the interior of S, i.e., we can always find a subgradient (on the boundary of the set S some difficulties may occur due, e.g., to discontinuities, but we need not be concerned with this technicality in the following). Then, the stationarity condition may be extended to 0 f (x ).

29 Unconstrained nonlinear programming Kelley s cutting planes Penalty function methods Lagrange multipliers and KKT conditions OVERVIEW The generic constrained problem min x S f (x) is normally stated in more concrete terms of equality and inequality constraints as: min f (x) s.t. h i (x) = 0, i E g i (x) 0, x X R n i I The condition x X may include additional restrictions, such as integrality of some decision variables, which destroy the convexity of S. We have a convex problem if both f ( ) and S are convex. As a general rule, convex problems are relatively easy (no trouble with local optima). There is a wide array of methods depending on the nature of objective function and constraints.

30 Unconstrained nonlinear programming Kelley s cutting planes Penalty function methods Lagrange multipliers and KKT conditions OVERVIEW Unconstrained optimization Penalty function methods Lagrange multipliers and KKT conditions (continuous case) Duality: decomposition and primal dual methods

31 Unconstrained nonlinear programming Kelley s cutting planes Penalty function methods Lagrange multipliers and KKT conditions UNCONSTRAINED OPTIMIZATION To solve the unconstrained problem min x R n f (x), we may use derivative-based methods such as: Steepest descent x (k+1) = x (k) α (k) f (x (k) ) f (x (k) ), for some step-size α (k) (difficulties in convergence, zig-zagging). Newton method, relying on a second-order local model of the objective to find the displacement δ: f (x (k) + δ) f (x (k) ) + [ f (x (k) )] T δ δt H(x (k) )δ, where H is the Hessian matrix. If H is positive definite, find a minimizer for the quadratic approximation by solving the system of linear equations H(x (k) )δ = f (x (k) ) and then set x (k+1) = x (k) + δ

32 Unconstrained nonlinear programming Kelley s cutting planes Penalty function methods Lagrange multipliers and KKT conditions UNCONSTRAINED OPTIMIZATION Several variants such as quasi-newton and trust-region; often finite differences are used to approximate derivatives. If the function is non-differentiable but convex, a subgradient method and its variants can be applied. However, sometimes the function is just a black box, as in stochastic simulation-based optimization. A host of derivative-free methods are available: simplex search (not to be confused with the simplex method for LP), pattern search, genetic algorithms (OK for nonconvex case), particle swarm optimization (OK for nonconvex case). Derivative-free methods do not even require continuity, but there are intermediate cases, where we do not know the form of the function, but we may find its value and a subgradient at any given point.

33 Unconstrained nonlinear programming Kelley s cutting planes Penalty function methods Lagrange multipliers and KKT conditions KELLEY S CUTTING PLANES Consider the convex problem min x S f (x), where the objective function f is actually not known in analytical form. Suppose that, for a given point x k, we are not only able to compute the function value f (x k ) = α k, but also a subgradient γ k, which does exist if the function is convex on the set S. In other words, we are able to find an affine function such that f (x k ) = α k + γ T k xk (4) f (x) α k + γ T k x x S. (5) In stochastic programming with recourse, we can do so for the recourse function.

34 Unconstrained nonlinear programming Kelley s cutting planes Penalty function methods Lagrange multipliers and KKT conditions KELLEY S CUTTING PLANES The availability of such a support hyperplane suggests the possibility of approximating f from below, by the upper envelope of support hyperplanes, f(x) x The Kelley s cutting plane algorithm exploits this idea by building and improving a lower bounding function until some convergence criterion is met. If S is polyhedral, we solve a sequence of LPs.

35 Unconstrained nonlinear programming Kelley s cutting planes Penalty function methods Lagrange multipliers and KKT conditions KELLEY S CUTTING PLANES STEP 0. Let x 1 S be an initial feasible solution; initialize the iteration counter k 0, the upper bound u 0 = f (x 1 ), the lower bound l 0 =, and the lower bounding function β 0 (x) =. STEP 1. Increment the iteration counter k k + 1. Find a subgradient of f at x k, such that equation (4) and condition (5) hold. STEP 2. Update the upper bound u k = min{u k 1, f (x k )} and the lower bounding function β k (x) = max{β k 1 (x), α k + γ T k x}. STEP 3. Solve the problem l k = min x S β k (x), and let x k+1 be the optimal solution. STEP 4. If u k l k < ɛ, stop: x k+1 is a satisfactory approximation of the optimal solution; otherwise, go to step 1.

36 Unconstrained nonlinear programming Kelley s cutting planes Penalty function methods Lagrange multipliers and KKT conditions PENALTY FUNCTIONS The problem with equality constraints min f (x) s.t. h i (x) = 0, i E can be approximated by the unconstrained problem min Φ(x, σ) = f (x) + σ X hi 2 (x). i E If σ is large enough, the optimization algorithm will, in some sense, first drive the solution toward the feasible region by minimizing the penalty term; then it will try to minimize the objective f. Actually, convergence difficulties will arise if we try solving the unconstrained problem with a large value of the penalty coefficient σ. So it is advisable to solve a sequence of unconstrained problems using the optimal solution of each subproblem as the initial solution of the next one.

37 Unconstrained nonlinear programming Kelley s cutting planes Penalty function methods Lagrange multipliers and KKT conditions PENALTY FUNCTIONS In the case of inequality constraints min f (x) s.t. g i (x) 0 i I, we must only penalize positive values of the constraint functions g i. Using the notation y + = max{y, 0}, we may use a penalty function like f (x) + σ X i I ˆg+ i (x) 2 or f (x) + σ X i I g + i (x) for increasing values of σ.

38 Unconstrained nonlinear programming Kelley s cutting planes Penalty function methods Lagrange multipliers and KKT conditions INTERIOR VS. EXTERIOR PENALTIES So far, we have seen exterior penalty functions. The feasible set is approached from outside for increasing values of the penalty coefficient σ. If the optimal solution is on the boundary of the feasible set (which is usually the case, since some inequality constraints are active), a feasible solution is obtained only in the limit. In some cases, this is quite natural, as the constraints may be soft or elastic, expressing some desirable feature rather than a hard requirement. In other cases, we would like to be able to stop the algorithm whenever we want and still come up with a strictly feasible solution. To overcome this problem, an interior penalty approach can be pursued.

39 Unconstrained nonlinear programming Kelley s cutting planes Penalty function methods Lagrange multipliers and KKT conditions INTERIOR VS. EXTERIOR PENALTIES P(x) x 2 S (exterior penalty) increasing g(x) x 1 (interior penalty) P(x) x 2 S decreasing g(x) x 1

40 Unconstrained nonlinear programming Kelley s cutting planes Penalty function methods Lagrange multipliers and KKT conditions BARRIER FUNCTIONS Interior penalty methods are based on a suitable barrier function. One example is B(x) = X i I 1 g i (x). The barrier function goes to infinity when x tends to the boundary of the feasible region from inside. Then an unconstrained problem, min f (x) + σb(x), is solved for decreasing values of σ, until the term σb(x) is small enough. An alternative is the logarithmic barrier function: B(x) = X i I log( g i (x)).

41 Unconstrained nonlinear programming Kelley s cutting planes Penalty function methods Lagrange multipliers and KKT conditions CLASSICAL LAGRANGE MULTIPLIERS Given the equality constrained case: min f (x) (6) s.t. h j (x) = 0, j = 1,..., m a necessary condition for local optimality of x is that there exist numbers λ j, j = 1,..., m, called Lagrange multipliers, such that f (x ) + mx λ j h j (x ) = 0 j=1 Note that the theorem is somewhat weak, in that it gives a necessary condition for local optimality, assuming differentiability and some regularity condition on constraints.

42 Unconstrained nonlinear programming Kelley s cutting planes Penalty function methods Lagrange multipliers and KKT conditions CONSTRAINT QUALIFICATION Consider the problem min x 1 + x 2 s.t. h 1 (x) = x 2 x1 3 = 0 h 2 (x) = x 2 = 0 and build the Lagrangian function L(x 1, x 2, λ 1, λ 2 ) = x 1 + x 2 + λ 1 (x 2 x1 3 ) + λ 2 x 2 The stationarity conditions yield the system L = 1 3λ 1 x1 2 = 0 x 1 L = x 2 x1 3 = 0 λ 1 This system of equations has no solution. L = 1 + λ 1 + λ 2 = 0 x 2 L = x 2 = 0 λ 2

43 Unconstrained nonlinear programming Kelley s cutting planes Penalty function methods Lagrange multipliers and KKT conditions CONSTRAINT QUALIFICATION In the example, the feasible set boils down to (0, 0), which is the (trivial) optimal solution. Unfortunately, the gradients of the two constraints are parallel at the origin and they are not a basis able to express the gradient of f. There are different constraint qualification conditions that may be applied (see Bazaraa et al.), such as: the gradients of functions h j are linearly independent at x ; constraints are linear; the candidate point is an interior (Slater) point (this applies only to inequality constraints; a more precise statement is needed for equality constraints).

44 Unconstrained nonlinear programming Kelley s cutting planes Penalty function methods Lagrange multipliers and KKT conditions KARUSH-KUHN-TUCKER CONDITIONS Consider a general constrained problem (P EI ) min f (x) s.t. h i (x) = 0, i E g i (x) 0, i I, and build the Lagrangian function L(x, λ, µ) = f (x) + X i E λ i h i (x) + X i I µ i g i (x). (7) Then (subject to the aforementioned conditions) a necessary condition for the local optimality of x is that there exist numbers λ i (i E) and µ i 0 (i I) such that f (x ) + X λ i h i (x ) + X µ i g i (x ) = 0 i E i I µ i g i (x ) = 0, i I.

45 Unconstrained nonlinear programming Kelley s cutting planes Penalty function methods Lagrange multipliers and KKT conditions KARUSH-KUHN-TUCKER CONDITIONS The KKT conditions are similar to classical Lagrange theorem, with two differences 1 Multipliers associated with inequality constraints are restricted in sign. 2 There is an additional condition, known as complementary slackness. If we interpret Lagrange multipliers economically, as shadow prices for resources, we may understand the two conditions: Prices cannot be negative. If a resource is not used to the limit and the associated constraint is not active, i.e., g i (x ) < 0, its shadow price must be zero. If the shadow price is positive, then the resource budget constraint must be active, i.e., g i (x ) = 0.

46 Unconstrained nonlinear programming Kelley s cutting planes Penalty function methods Lagrange multipliers and KKT conditions MULTIPLIERS METHODS From an algorithmic point of view, the KKT conditions are not solved directly. They are the conceptual basis for computationally viable methods. There are methods integrating multipliers with penalty functions, such as augmented Lagrangians. They are fundamental in decomposition strategies. For that, we need duality theory.

47 The role of duality Weak duality Strong duality Dual decomposition THE ROLE OF DUALITY Duality is essential in developing solution methods. Dual simplex (essential in integer programming). Primal-dual methods, like interior point methods. Decomposition methods. Duality has a useful economic interpretation.

48 The role of duality Weak duality Strong duality Dual decomposition WEAK DUALITY Consider the inequality-constrained problem (P) min f (x) This problem is called the primal problem. s.t. g i (x) 0 i I (8) x S R n. The set S is any subset of R n, possibly a discrete one. Here, we do not assume differentiability nor convexity of the objective function. The results we get are therefore extremely general. Build the Lagrangian function by dualizing constraints (8): L(x, µ) = f (x) + X i I µ i g i (x) = f (x) + µ T g(x). For a given multiplier vector µ (dual variables), the minimization of the Lagrangian function with respect to x S is called the relaxed problem.

49 The role of duality Weak duality Strong duality Dual decomposition WEAK DUALITY The approach makes sense when the inequality constraints are the complicating ones, and the relaxed problem is easy to solve. The solution of the relaxed problem defines a function w(µ), called the dual function: w(µ) = min L(x, µ). x S Consider the dual problem: (D) max w(µ) = max µ 0 µ 0 j ff min L(x, µ). (9) x S (Weak duality theorem) For any µ 0, the dual function is a lower bound for the optimum f (x ) of the primal problem (P), i.e., w(µ) f (x ) µ 0.

50 The role of duality Weak duality Strong duality Dual decomposition WEAK DUALITY Proof. Let us adopt the notation ν(p) to denote the optimal value of the objective function for an optimization problem P. Under the hypothesis µ 0, it is easy to see that 0 1 min ν(p) s.t. 0 min s.t. f (x) x S µ T g(x) 0 min f (x) + µ T g(x) ν s.t. x S A (10) 1 f (x) + µ T g(x) x S A (11) µ T g(x) 0 «. (12)

51 The role of duality Weak duality Strong duality Dual decomposition WEAK DUALITY The case of equality constraints is treated similarly: in that case, multipliers are unrestricted in sign. We obtain a very general but weak relationship, since weak duality only yields a lower bound. The following theorem gives a sufficient condition for global optimality: THEOREM If there is a pair (x, µ ), where x S and µ 0, satisfying the conditions 1 f (x ) + (µ ) T g(x ) = min x S {f (x) + (µ ) T g(x)}, 2 (µ ) T g(x ) = 0, 3 g(x ) 0, then x is a global optimum for the primal problem (P).

52 The role of duality Weak duality Strong duality Dual decomposition STRONG DUALITY The last theorem is, in a sense, stronger than necessary. Luckily, there are cases in which we get necessary and sufficient conditions. Indeed, under suitable conditions (essentially convexity), a stronger property holds, known as strong duality: ν(d) = w(µ ) = f (x ) = ν(p). The convexity assumption does not hold, in particular, for the case of a discrete set and for general equality constraints (unless they are affine functions). To be precise, convexity is not enough: we may have a duality gap unless additional conditions, such as the Slater constraint qualification, hold.

53 The role of duality Weak duality Strong duality Dual decomposition STRONG DUALITY: COMPUTATIONAL APPROACH In principle, strong duality provides us with an alternative approach to solve the primal problem. This makes sense when we may get rid of complicating constraints by dualization. However, we need a computational scheme: 1 Assign an initial value µ (0) 0; set k 0. 2 Solve the relaxed problem with multipliers µ (k). 3 Given the solution ˆx (k) of the relaxed problem, compute a search direction s (k) and a step length α (k), and update the multipliers (making sure they stay non-negative): n µ (k+1) = max 0, µ (k) + α (k) s (k)o. Then set k k + 1, and go to step 2.

54 The role of duality Weak duality Strong duality Dual decomposition STRONG DUALITY: COMPUTATIONAL APPROACH In order to find a search direction, one would be tempted to compute a gradient of the dual function. Unfortunately, the dual function need not be everywhere differentiable, but the following facts help us: THEOREM The dual function w(µ) is a concave function. THEOREM Let ˆx be an optimal solution of the relaxed problem for a multiplier vector ˆµ. Then g(ˆx) is a subgradient of the dual function at ˆµ.

55 The role of duality Weak duality Strong duality Dual decomposition DUAL DECOMPOSITION To see how duality can help in a simple setting, let us consider a problem like nx max f i (x i ) (13) s.t. i=1 nx g i (x i ) b, (14) i=1 x i S i, i = 1,..., n (15) Let us interpret the decision variables x i, i = 1,..., n, as activities yielding a profit f i (x i ) and consuming a resource amount g i (x i ). The objective function (13) is total profit, and (14) is a budget constraint on the resource. Note that the objective function is measured in monetary terms, whereas b is measured in resource units. If we could get rid of the budget constraint, the problem could be decomposed.

56 The role of duality Weak duality Strong duality Dual decomposition DUAL DECOMPOSITION Let us dualize the budget constraint by introducing the multiplier µ 0 and writing the Lagrangian function:! nx nx nx L(x, µ) = f i (x i ) + µ b g i (x i ) = [f i (x i ) µg i (x i )] + µb. i=1 i=1 Note that here we must adjust the problem to account for the optimization sense. The Lagrangian function should be maximized with respect to the primal variable, resulting in a set of problems: i=1 max [f i (x i ) µg i (x i )] π i (x i ) x i S i

57 The role of duality Weak duality Strong duality Dual decomposition DUAL DECOMPOSITION Each subproblem requires maximizing profit contribution minus resource cost. The multiplier µ is a shadow price, measured in unit of money per unit amount of resource. In this case, we should minimize the dual function with respect to µ. Given relaxed solutions x i, we get a subgradient of the dual function: nx g i (x i ) b i=1 This is positive when the budget is exceeded, in which case we should increase the resource price. The price must be reduced when the budget is not exceeded. The resulting demand-offer scheme can be depicted as follows.

58 The role of duality Weak duality Strong duality Dual decomposition DUAL DECOMPOSITION min w( ) x 1 x n max 1(x 1) max n(x n) x 1 x n

59 The role of duality Weak duality Strong duality Dual decomposition DUAL DECOMPOSITION Dual decomposition may converge poorly in practice, but it might be a good approach for some specially structured large-scale problems. Sometimes, we are satisfied by a suitably food solution. If we may recover a good primal feasible solution from dual decomposition, we obtain dual heuristic algorithm. Lagrangian methods can be integrated with penalty function methods, resulting in augmented Lagrangian schemes based, e.g., on the minimization of f (x) + X λ i h i (x) + σ X hi 2 (x) i I i I for an equality-constrained problem.

60 Simplex method for linear programming Duality in linear programming An interior point method for LP LINEAR PROGRAMMING: SIMPLEX METHOD Extremely efficient and robust approach to solve linear programs in the form min s.t. c T x Ax = b x 0 where x R n, c R n, A R m,n, and b R m. The idea relies on the fact that there is an optimal solution corresponding to a vertex of the polyhedron. Given a vertex, we can explore neighboring vertices to see if there is a better one and move there. Since the problem is convex, we end up in the global solution. The geometric intuition must be translated into algebraic terms. The problem requires to express b by an optimal conic combination of columns of A.

61 Simplex method for linear programming Duality in linear programming An interior point method for LP LINEAR PROGRAMMING: SIMPLEX METHOD There are n columns, but a subset B of m columns suffices to express b as follows: X a j x j = b j B To be precise, we should make sure that this subset of columns is a basis, i.e., that they are linearly independent; let us cut a few corners and assume that this is the case. A solution of this system, in which n m variables are set to zero, and only m are allowed to assume a nonzero value is a basic solution. A basic feasible solution corresponds to an extreme point of the feasible set: hence, moving from one vertex to a neighboring one is obtained by bringing one column into the basis, eliminating one column. There are some computational issues to be tackled, related to degeneracy and redundant constraints and to efficient implementation. Alternative interior-point methods move in the interior of the polyhedron.

62 Simplex method for linear programming Duality in linear programming An interior point method for LP DUALITY IN LINEAR PROGRAMMING Duality in LP can be derived from first principles of convexity (separation theorems, Farkas lemma, and the like); however, it may be preferable to cast it in the more general, nonlinear framework. Let us start with an LP problem (P 1 ) in the following canonical form: (P 1 ) min c T x s.t. Ax b. If we dualize the inequality constraints, we get the dual problem: n o n o max min c T x + µ T (b Ax) = max µ T b + min c T µ T A x. µ 0 x µ 0 x Since x is unrestricted in sign, the inner minimization problem has a finite value if and only if c T µ T A = 0.

63 Simplex method for linear programming Duality in linear programming An interior point method for LP DUALITY IN LINEAR PROGRAMMING Since we want to maximize the dual function, we enforce the condition, and the dual problem (D 1 ) turns out to be (D 1 ) max b T µ s.t. A T µ = c µ 0. The dual problem is still an LP problem, resulting from: 1 the exchange of b with c, 2 the transposition of A, 3 a change in the sense of the objective. Using the same reasoning, we may build the dual of an arbitrary LP problem. Note, in particular, that dual variables associated with equality constraints will be unrestricted in sign.

64 Simplex method for linear programming Duality in linear programming An interior point method for LP DUALITY IN LINEAR PROGRAMMING Since there is no duality gap, if both primal and dual problems have a finite optimum, we have b T µ = c T x However other cases are possible: the primal is unbounded below, and the dual is infeasible; the dual is unbounded above, and the primal is infeasible; both problems are infeasible. Information on dual variables is provided by the primal simplex method. When an LP problem is unbounded, the simplex algorithm returns a recession direction, i.e., a direction on which we may move and improve the objective, without ever going infeasible.

65 Simplex method for linear programming Duality in linear programming An interior point method for LP PRIMAL-DUAL BARRIER METHOD FOR LP By combining several principles, we obtain an interior point method for LP Let us consider the primal problem max s.t. c T x Ax b x 0 max s.t. c T x Ax + w = b x, w 0 and its dual: min s.t. b T y A T y c y 0 where w and z are slack variables. min s.t. b T y A T y z = c y, z 0

66 Simplex method for linear programming Duality in linear programming An interior point method for LP PRIMAL-DUAL BARRIER METHOD FOR LP We can get rid of non-negativity constraints by using an interior penalty function based on a logarithmic barrier. max c T x + σ X j log x j + σ X i log w i s.t. Ax + w = b. Equality constraints can be dualized by Lagrange multipliers y, yielding the Lagrangian function L(x, w, y) = c T x + σ X j log x j + σ X i log w i + y T (b Ax w).

67 Simplex method for linear programming Duality in linear programming An interior point method for LP PRIMAL-DUAL BARRIER METHOD FOR LP We can now apply the first-order stationarity conditions on the Lagrangian. L = c j + σ 1 X y i a ij = 0, x j x j i L = σ 1 y i = 0, i w i w i L = b i X a ij x j w i = 0, y i j j i. These optimality equations may be rewritten in a compact matrix form: x 1 A T y σx 1 e = c y = σw 1 x 2 e where X = , e = 6 4 Ax + w = b. xn

68 Simplex method for linear programming Duality in linear programming An interior point method for LP PRIMAL-DUAL BARRIER METHOD FOR LP To make the meaning of the optimality conditions clearer, let us introduce the auxiliary vector z = σx 1 e and let us rearrange the conditions as: Ax + w = b A T y z = c XZe = σe YWe = σe. These equations have a nice interpretation in terms of (see Theorem 1). 1 primal feasibility, 2 dual feasibility, 3 and (if σ = 0) complementary slackness.

69 Simplex method for linear programming Duality in linear programming An interior point method for LP PRIMAL-DUAL BARRIER METHOD FOR LP For σ > 0, we have a set of nonlinear equations: 2 F(ξ) = 0, where ξ = which may be tackled by Newton s method. In principle, by solving this system of non-linear equations for different values of σ we get a path (x σ, y σ, w σ, z σ). This path is called central path and for 6 4 x y w z 3 7 5, σ 0, it leads to the optimal solution of the original LP.

70 Simplex method for linear programming Duality in linear programming An interior point method for LP PRIMAL-DUAL BARRIER METHOD FOR LP The method can be interpreted as an instance of more general homotopy-based strategies. Essentially, we keep a pair of primal and dual feasible solutions and we gradually enforce complementary slackness. There are different approaches to manage the interplay between Newton steps and adjusting the penalty parameter. The Newton steps in interior point methods can be more or less efficient, as they require solving large-scale systems of linear equations (sparsity of Cholesky factors is essential).

71 Simplex method for linear programming Duality in linear programming An interior point method for LP SIMPLEX VS. INTERIOR-POINT METHODS In the worst-case, the simplex method has exponential complexity, whereas interior-point methods are polynomial. We may get equivalent, but different solutions, as interior point methods tend to yield solutions in the midpoint of a face of the polyhedron, whereas simplex yields an extreme point (crossover to a basic solution is offered after solution with interior point methods). In practice, no method dominates, as behavior may depend on problem structure. Simplex has better warm-start capabilities (useful for integer programming). Interior-point methods have been extended to a much wider class of convex optimization problems, including second-order cone and semidefinite programming.

72 Conic programming Second order cone programming Semidefinite programming Conic duality CONIC PROGRAMMING is an important case of convex optimization. The next level in the hierarchy is a quadratic programming (QP) problem like: min s.t. 1 2 xt Qx + h T x Ax = b x 0 This is a convex optimization problem, provided that Q S n +. Interior-point methods are available for QP as well, but they are also available for a wider class of problems.

73 Conic programming Second order cone programming Semidefinite programming Conic duality CONIC PROGRAMMING Both LPs and QPs can be reformulated within a wider class of conic problems: Let V and W be (finite-dimensional) linear spaces equipped with an inner product. Let K V and L W be closed convex cones. Let A : V W be a linear operator. A conic program is an optimization problem like where c V and b W. min c, x s.t. b A(x) L x K

74 Conic programming Second order cone programming Semidefinite programming Conic duality SECOND ORDER CONE PROGRAMMING Second order cone programming problems (SOCP) are a generalization of LP and QP: min f T x s.t. A i x + b i 2 c T i x + d i, i = 1,..., n Fx = g A constraint like Ax + b 2 c T x + d requires that the affine mapping of point x R n to (Ax + b, c T x + d), where A R k,n and b R k, lies in the second-order (Lorentz) cone in R k+1.

75 Conic programming Second order cone programming Semidefinite programming Conic duality SEMIDEFINITE PROGRAMMING In semidefinite programming, decision variables are symmetric, semidefinite positive matrices X S n +. On the self-dual cone S n + we define the inner product nx X, Y tr(xy) = x ij y ij X Y i,j=1 (in the more general case, the first matrix should be transposed). Then we may formulate a model like max C X (16) s.t. A 1 X = b 1. A m X = b m X 0

76 Conic programming Second order cone programming Semidefinite programming Conic duality CONIC DUALITY SOCPs and SDPs have wide applicability, including some stochastic and robust optimization problems. Rather efficient interior-point methods are available for their solution, based on the following conic duality theorem. Consider problems: (P) max c, x (D) min b, y s.t. b A(x) L s.t. A T (y) c K x K where A T ( ) is the adjoint operator of A( ). y L

77 Conic programming Second order cone programming Semidefinite programming Conic duality CONIC DUALITY The adjoint operator is a linear operator A T : W V such that y, A(x) W = A T (y), x V, for every x V and y W. When dealing with matrices, i.e., linear mappings between usual vector spaces R m and R n, the adjoint boils down to the familiar transpose of a matrix. THEOREM If the primal problem (P) is feasible, has a finite value γ, and has an interior (Slater) point ˆx, then the dual problem (D) is also feasible and has the same value γ. The Slater constraint qualification condition is necessary, in general, to rule out some pathological cases resulting in duality gaps.

78 Conic programming Second order cone programming Semidefinite programming Conic duality CONIC DUALITY: THE SDP CASE Strong duality applies to the SDP (16), since the cone S n + of semidefinite positive matrices has an interior, the open cone S n ++ of positive definite matrices. To be precise, there must be a symmetric positive definite matrix ˆX such that the equality constraints hold, A(ˆX) = b, where b = [b 1,..., b m] T, and A( ) collects the matrices A i and maps S n + to R m. The dual of problem (16) is: min s.t. b T y mx y i A i C 0, i=1 y R m To devise primal-dual interior point methods, a barrier function is needed. A possible choice is log(det(x)).

79 Conic programming Second order cone programming Semidefinite programming Conic duality CONIC DUALITY: A NUMERICAL EXAMPLE Consider the following symmetric matrix: A = The matrix is not defined in sign, as its eigenvalues are , , We want to find a (symmetric) matrix X 0 with minimal trace (sum of eigenvalues) such that X A 0.

80 Conic programming Second order cone programming Semidefinite programming Conic duality CONIC DUALITY: A NUMERICAL EXAMPLE Actually, we can solve the problem by diagonalizing A = VΛV T, where Λ is a diagonal matrix consisting of the eigenvalues of A, and the columns of V are the corresponding (unit) eigenvectors. Then we collect the positive eigenvalues of A into Λ + and set X = VΛ +V T. Using SDP, we may solve: min tr(x) = I X s.t. X A 0 X 0, or its dual (if we just care about the objective): max A Y s.t. I Y 0 Y 0.

81 Conic programming Second order cone programming Semidefinite programming Conic duality HOMEWORK For Ph.D. students who formally need credits: 1 Using conic duality, find the dual of the LP problem in standard form: min s.t. c T x Ax = b x 0