Duality dual problem

Linear Programming.28—A linear programming problem as defined in matrix notation requires that a vector x 0 (non-negativity constraints) be found that satisfies the constraints Ax <, b, and maximizes the linear function cx. Here x = (xx, , xn), A = [aiy] (i = 1,- -,m j = 1,- , ), b - (61 - -,bm), and c = (cu- -,c ) is the cost vector. With the original (the primal) problem is associated the dual problem yA > c, y > 0, bij = minimum, where y yx,- , ym)-A duality theorem 29 asserts that if either the primal or the dual has a solution then the values of the objective functions of both problems at the optimum are the same. It is a relatively easy matter to obtain the solution vector of one problem from that of the other. 

Nonlinear optimization problems have two different representations, the primal problem and the dual problem. The relation between the primal and the dual problem is provided by an elegant duality theory. This chapter presents the basics of duality theory. Section 4.1 discusses the primal problem and the perturbation function. Section 4.2 presents the dual problem. Section 4.3 discusses the weak and strong duality theorems, while section 4.4 discusses the duality gap. 

Remark 2 It is not always possible to obtain the optimal value of the dual problem being equal to the optimal value of the primal problem. This is due to the form that the image set I can take for different classes of mathematical problems (i.e., form of objective function and constraints). This serves as a motivation for the weak and strong duality theorems to be presented in the following section. 

In the previous section we have discussed geometrically the nature of the primal and dual problems. In this section, we will present the weak and strong duality theorems that provide the relationship between the primal and dual problem. 

Remark 4 This important lower-upper bound result for the dual-primal problems that is provided by the weak duality theorem, is not based on any convexity assumption. Hence, it is of great use for nonconvex optimization problems as long as the dual problem can be solved efficiently. 

Remark 1 Result (ii) precludes the existence of a gap between the primal problem and dual problem values which is denoted as duality gap. It is important to note that nonexistence of duality gap is guaranteed under the assumptions of convexity of f(x))g(x), affinity of h(x), and stability of the primal problem (P). 

Remark 6 The geometrical interpretation of the primal and dual problems clarifies the weak and strong duality theorems. More specifically, in the vicinity of y — 0, the perturbation function v(y) becomes the 23-ordinate of the image set I when zi and z2 equal y. In Figure 4.1, this ordinate does not decrease infinitely steeply as y deviates from zero. The slope of the supporting hyperplane to the image set I at the point P, (-pi, -p2), corresponds to the subgradient of the perturbation function u(y) at y = 0. 

Part 1, comprised of three chapters, focuses on the fundamentals of convex analysis and nonlinear optimization. Chapter 2 discusses the key elements of convex analysis (i.e., convex sets, convex and concave functions, and generalizations of convex and concave functions), which are very important in the study of nonlinear optimization problems. Chapter 3 presents the first and second order optimality conditions for unconstrained and constrained nonlinear optimization. Chapter 4 introduces the basics of duality theory (i.e., the primal problem, the perturbation function, and the dual problem) and presents the weak and strong duality theorem along with the duality gap. Part 1 outlines the basic notions of nonlinear optimization and prepares the reader for Part 2. 

The term duality is often used to invoke a contrast between two related concepts. Duality is one of the most fundamental concepts in mathematical programming and establishes a connection between two symmetric programs, namely, the primal and dual problem. 

Theorem A.2 Weak duality For any feasible solution x of the primal problem (A.l) and for any feasible solution X, fi, of the dual problem (A.8), the following holds... 

Feasible x) and y) give upper and lower bounds on the optimal value of the objective function, which in the 2-RDM problem is the ground-state energy in a finite basis set. The primal and dual solutions, x) and y), sie feasible if they satisfy the primal and dual constraints in Eqs. (107) and (108), respectively. The difference between the feasible primal and the dual objective values, called the duality gap fi, which equals the inner product of the vectors x) and z). 

The weak duality theorem provides the lower-upper bound relationship between the dual and the primal problem. The conditions needed so as to attain equality between the dual and primal solutions are provided by the following strong duality theorem. 

The derivation of the Lagrange relaxation master problem employs Lagrangian duality and considers the dualization of the i(ac,y) < 0 constraints only. The dual takes the following form ... 

We have generally used the direct construction of an objective function as some measure of the profit. This then has to be maximized. An important form of problem is to take the objective function as a measure of the cost and to minimize it. In some cases the same problem can be formulated in two ways which are duals of one another. Thus, if we seek the minimum holding time to achieve a given conversion we are solving the dual of the problem of finding the maximum conversion for given holding time. The existence of duality is useful but it needs to be carefully established, as, for example, in Amundson and Bilous (1956). 

The problem with this particular duality is that the dual of an inflexion in a curve is an asymptote, so that if the polygon has an inflexion the curve will contain pieces of hyperbola going off to infinity. If rules are put in place that something special is done when the polygon has an inflexion, however, this can give a nice shape-preserving scheme which has no inflexions in the limit curve when the initial polygon has none. 

So does that mean we need to jettison our precious duality principle altogether Actually we don t. The problem here is that the two circuits in Figure 1-3, despite being deceptively similar, are really not duals of each other. And for that reason, we really can t use them to derive any clues either. A little later, we will construct proper dual circuits. But for now we may have already started to suspect that we really don t understand inductors as well as we thought, nor in fact the duality principle we were perhaps counting on to do so. 

More research efforts are required to address decentralized SCs problems. As previously mentioned, duality, and separability principles may provide frameworks to achieve overall optimal solutions by interchanging among SC partners noncrit-ical information (i.e., dual values) instead of usually confidential data (e.g., costs, prices, technology parameters). Additionally, it would be interesting to explore the use of complementarity programs for decentralized SC modeling. 

For nonconvex programs, the difference between the optimal objective function values of the dual and primal problems ((p (X, p )-f(x ))is called duality gap. For nonconvex programs of engineering applications, the duality gap is usually relatively small (Conejo et al. 2002). 

Lagrangian relaxation is a technique that is suitable for problems with complicating constraints. The idea is to apply the duality function (see Sect. A. 1.3) to this kind of problems in order to reduce their complexity (Guignard 2003). At this point, it is noteworthy that not all the problem constraints must be included in the Lagrangian function in order to construct the dual function (Bazaraa et al. 1993). The Lagrangian... 

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