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2. Fundamental Idea Assume the issue gets to be raised when certain variables are altered. On the off chance that these variables are discrete, we can reformulate the issue as disjunctions of curved limitations. On the off chance that some of them are consistent, discretize them to acquire an inexact worldwide arrangement. Inspiration is to exploit propelled arrangement techniques: Branch-and-bound system picks the fitting disjunct in eac

Rationale Based Methods for Global Optimization J. N. Hooker Carnegie Mellon University, USA November 2003

Basic Idea Assume the issue gets to be distinctly arched when certain factors are settled. On the off chance that these factors are discrete, we can reformulate the issue as disjunctions of curved imperatives. In the event that some of them are persistent, discretize them to get an inexact worldwide arrangement. Inspiration is to exploit propelled arrangement techniques: Branch-and-bound strategy picks the proper disjunct in every limitation. Nonlinear programming technique fathoms raised subproblems that outcome when disjuncts are picked.

Outline General type of issue Structural plan illustration Disjunctive definition Branch and bound with arched relaxations Big-M detailing Convex body detailing Logic-based external guess Logic-based Benders deterioration Branch and bound with curved semi relaxations Realistic basic outline issue Solution by MILP Solution by semi unwinding Other applications

If is ceaseless, discretize it, to get inexact worldwide arrangement. General Form of Problem Vector of capacities Logical conditions on y Assume that when is settled to , we get a curved issue arched elements of x

Objective is characterized in the limitations We accept one for each requirement. Numerous issues have this shape. If not, requirements can on a fundamental level be put into this shape by change of variable.

For instance, consider Use the change of variable and the imperatives have the wanted frame: One y j for each limitation Logical condition

stack = 10 relocation = uprooting = stack = 20 cost of steel punishment for dislodging Structural Design Example Choose bar thickness that limits cost. This case is planned just to represent the calculations. A more practical model for auxiliary plan is exhibited toward the finish of the discussion. = pressure of bar j = thickness (cross-sectional zone) of bar j cost =

Can be composed in fancied frame: Global enhancement issue: Hooke's law dislodge ment or many firmly divided qualities for nonstop issue thickness

How to Solve It? Can on a basic level utilize branch and bound by stretching on . Be that as it may, nonstop relaxations at hubs of the pursuit tree are by and large nonconvex . So we compose the issue in disjunctive shape.

Disjunctive Formulation Now each disjunct is raised. We will explain by: Branch and bound with arched relaxations (utilize disjunctive programming or MINLP) . Rationale based external estimation with direct relaxations. Relaxations can be substantial when there are numerous disjunctions. For this situation consider: Logic-based Benders disintegration with discrete unwinding. Branch and bound with arched semi relaxations (requires that imperative capacities fulfill certain properties).

Recall the illustration… Disjunctive definition is: Disjuncts are raised (straight)

Branch and Bound with Convex Relaxations Two curved relaxations of a disjunction Big-M : Write a major M detailing with 0-1 factors and take its persistent unwinding (i.e., drop the integrality prerequisite on the 0-1 factors). Arched frame : Write a raised body definition with 0-1 factors and take its constant unwinding. Two arrangement alternatives Disjunctive programming : branch on disjunctions. Blended number nonlinear programming (MINLP): branch on 0-1 factors. Ideal estimation of unwinding gives a lower bound that is utilized as a part of a branch-and-bound plan.

Big-M definition of disjunction The disjunction: Big-M plan: Where M v is a vector of legitimate upper limits on the part elements of g ( x,v ). It is expected that x is limited above and underneath. To acquire unwinding, supplant with

Relaxation… Projection is Example of huge M unwinding Disjunction…

Convex body definition of disjunction Stubbs & Mehrotra; Grossmann & Lee The disjunction: Assume every g ( x , v ) is limited and curved. Additionally expect x L x x U Write each point in the unwinding as a curved mix of focuses fulfilling the disjuncts Use change of variable Nonconvex

Restore convexity by increasing by raised This is an arched structure unwinding (i.e., ventures onto raised frame in x - space). Be that as it may, disaggregation of x includes numerous new factors. To get 0-1 definition, supplant with

Convex body unwinding… Example of curved frame unwinding Disjunction…

Solve auxiliary plan case with enormous M detailing Disjunctive detailing: Big-M detailing: Solve by disjunctive programming or MINLP. Get ideal arrangement at the root hub. In this manner

Solve basic plan case with arched frame definition Disjunctive detailing: Convex structure plan: Solve by disjunctive programming or MINLP.

Logic-Based Outer Approximation T ü rkay and Grossmann Allows one to utilize direct relaxations. In any case, one must explain a blended number straight programming (MILP) ace issue over and again. Take care of an ace issue containing 1 st - arrange approximations of the disjuncts to acquire an incentive for y . Understand with MILP, which utilizes straight relaxations. Fathom the subproblem that outcomes when y is settled to this esteem, to get an incentive for x . Figure 1 st - arrange approximations about beforehand acquired estimations of x, y . Proceed until estimation of ace issue best esteem got in a subproblem up until now. Start with warm begin by precomputing 1 st - arrange approximations around a few estimations of ( x,y ).

Disjunctive plan once more: The ace issue in cycle K + 1 is the place ( x k ,y k ) are arrangements from past emphasess. The nonlinear subproblem in cycle K is

Solve basic outline case with rationale based external estimation Disjunctive plan once more: Master issue: Disjuncts officially direct

Solve ace issue as MILP (Big-M formulation):` For warm begin, illuminate subproblem for 2 y 's: y 1 = (1,1), which yields x 1 = (20,20) y 2 = (2,2), which yields x 2 = (5,10). This outcomes in the ace issue:

Solve ace issue and get which suggests Subproblem arrangement is Next ace issue is new same Solution is and the calculation ends with y = (1,2).

Logic-Based Benders Decomposition Hooker and Ottosson Can be helpful when factors have an extensive number of discrete qualities, bringing about countless. Merging can be moderate. Take care of an ace issue for y . The ace issue deficiently depicts the projection of the first issue onto the y - space . Unravel the subproblem that outcomes when y is settled to this esteem. Get Benders cut from induction double of the subproblem . Add the slice to the ace issue to decide out a few arrangements that are no superior to the past one. Proceed until the ace and subproblem unite in esteem. Best to have a warm begin with "don't-be-idiotic" limitations including y .

Disjunctive detailing once more: The nonlinear subproblem in cycle K is Lagrange multiplier Optimal esteem = The ace issue in emphasis K + 1 is Logical Benders cuts

Solve basic outline case with rationale based Benders disintegration Initial ace issue: Don't-be-inept requirement One arrangement is Solve subproblem: Corresponds to y 1 = 1 Lagrange multipliers Corresponds to y 2 = 1 Subproblem arrangement is

Since the ace issue is Solution is Continue in this design. Ace issue in cycle 4 is: same Solution is The calculation ends with y = (1,2).

Branch and Bound with Convex Quasi-Relaxations Does not require disjunctive definition and is along these lines helpful when there are numerous discrete qualities. Be that as it may, the imperative capacities must have a specific frame. Take care of the issue by branch and bound. Acquire limits from semi relaxations at every hub. Given issue P : an issue Q : is a semi unwinding of P if for any achievable arrangement x of P , there is a possible arrangement x of Q with f ( x ) f ( x ) . Along these lines one can get a legitimate lower bound by tackling a semi unwinding.

Consider the issue, Theorem. Assume that each is either (a) curved [for ( i,j) J 1 ] or (b) inward in y j and homogeneous in x : [for ( i,j) J 2 ]. Assume likewise that Then the accompanying is a curved semi unwinding :

Why? Take any doable arrangement of To get an attainable arrangement of do the accompanying: concavity Then homogeneity

fulfilled, by development arched, on the grounds that g j ( x , y ) is curved fulfilled, by above contention fulfilled, by development raised, on the grounds that g j ( x , y ) is raised in x So we have a plausible arrangement of the semi unwinding with esteem that is not exactly or equivalent to (in reality equivalent to) that of the first issue.

Solve consistent adaptation of basic plan case with semi relaxations Original definition: Discretize Put in appropriate shape: arched Concave in y j & homogeneous in 1 st contention ( s j , x j )

The semi unwinding is: Can now re-total s j :

Beginning of branch-and-bound tree Total 63 hubs out of 31 31 conceivable arrangements. Get y = (1.1, 2.0) with z 0 = 1394.5 Root hub x 0 = 1177.8 = (0.667,0.667) y = (1,1) y 1 [0,1] y 1 [1.1,3] x 0 = 1322 = (0,0.667) y = (1,1) x 0 = 1283 = (0.816,0.667) y = (1.45,1) y 2 [1.1,3] y 2 [0,1] Global ideal is y = (1.126, 1.972) with z 0 = 1394.1 x 0 =1352 = (0,0.816) y = (1,1.45) x 0 =1900 = (0,0) y = (1,1) doable arrangement

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