decision modeling with microsoft excel copyright 2001 prentice hall publishers and ardith e. baker...
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DECISION MODELING WITH DECISION MODELING WITH MICROSOFT EXCELMICROSOFT EXCEL
Copyright 2001Prentice Hall Publishers and
Ardith E. Baker
NonlinearNonlinear
Chapter 7Chapter 7
OptimizationOptimizationPart 1Part 1
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Introduction to Nonlinear Introduction to Nonlinear Optimization ModelsOptimization Models
Not all ___________relationships in business and economics problems are _________.
In general, some of the prominent reasons for ________________are:
1. ________________Relationships
2. Nonadditive Relationships
3. Efficiencies or Inefficiencies of__________Although common, nonlinear models are
more difficult to ____________than linear models.
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Unconstrained Optimization in Unconstrained Optimization in Two Two
or More Decision Variablesor More Decision VariablesConsider the case of two ___________variables, x1 and x2 and a given function f(x1,x2).For the case of 2 decision variables, partial ____________from calculus are used to describe local or global ____________of f.
Let fxi denote the first partial derivative, fxi xi
denote the second partial derivative, etc.
Any point at which all first partial derivatives vanish is called a_____________________.
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We have the following ___________________for optimality:
At a local max or min both partial At a local max or min both partial derivatives must equal ______(i.e., derivatives must equal ______(i.e., f f xx11= = f f
xx22 = 0). = 0). That is, a local maximizer or a local That is, a local maximizer or a local
minimizer is always a _____________point.minimizer is always a _____________point.However, not all stationary points provide maxima and___________.
In this case, we can employ the second order ___________condition for optimality.
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First and second-order tests (called first-first-order ________conditionsorder ________conditions and ___________optimality conditionsoptimality conditions, respectively) can be applied to locate unconstrained local optima for functions of more than one variable.NOTES:
The first order conditions are____________; the second-order conditions are sufficient.The second-order conditions __________the first order ones (i.e., the second order conditions assume that x1
*,x2* is a ________ point).
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The first-derivative test (the necessary condition) says that the _____optima are contained among the stationary points of the__________.
The second-derivative test (the _________ condition) allows us to distinguish between local ___________and minimizers, and points that are neither.
For a ____________function of n variables, each local optimizer is a stationary point.
In order to guarantee that a stationary point is, for example, a _______maximizer, second-order sufficiency conditions must be___________.
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These two types of optimality conditions have limited practical ______________because:
1. Setting the first partial derivatives equal to zero gives a system of n equations in n ________. Unless the system is_____, it is not easy to find solutions.
2. The second-order sufficiency conditions are complicated and require the evaluation of _______________of certain entries in the matrix of second partial derivatives.
First-order necessary conditions in nonlinear optimizers serve as a _________criterioncriterion for the hill-climbing optimization methods that search for local__________.
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When maximizing a ___________functionfunction, any stationary point is a _________maximizer (for a convex function, any stationary point is a global minimizer).
In the general case, an optimized solution could be a local maximizer or minimizer or neither, in the _________case, we are guaranteed that any solution is a global_____________.
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Nonlinear Optimization: Nonlinear Optimization: A Descriptive Geometric A Descriptive Geometric
IntroductionIntroductionSoftware packages (such as Solver) are based on ________hill-climbing (or hill-descent) behavior. For a maximization problem:
An _________point is chosen (i.e., a set of numerical values for the decision variables).An uphill direction is determined by approximating the _________of change in all directions (the first partial derivative) of the objective function at that initial point.
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The method __________when the approximated rates of further OV change in all directions (the first ________derivatives) are close to zero (the first order conditions are__________).
For _____________optimization, the method moves from the initial point, along a line in an uphill (____increasing) direction, to the highest point that can be attained on that line.
Then, a new uphill direction is defined, and the ____________is continued.
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Although the focus has been on unconstrained ____________unconstrained ____________, we are interested in optimizing an objective function subject to______________. Just as in the case of LP modeling, constraints take the form of __________and/or inequalities. However, ____________of the constraints is not assumed in this case.
Thus, the general NLP ___________model can be written as follows (f and gi’s are just symbols for complicated nonlinear functions of the decision variables, x, to compute the OV and each constraint’s LHS).
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Max f(x1, x2, …, xn) (objective)
s.t. g1(x1, …, xn) = b1
g2(x1, …, xn) = b2
gm(x1, …, xn) = bm
m equalityconstraints
k inequalityconstraints
h1(x1, …, xn) < r1
h2(x1, …, xn) < r2
hk(x1, …, xn) < rk
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Nonlinear Optimization: Nonlinear Optimization: A Descriptive Geometric A Descriptive Geometric
IntroductionIntroduction
Consider the following symbolic model:Graphical Analysis:Graphical Analysis:
Max x1 - x2
s.t. -x12 + x2 > 1
x1 + x2 < 3
-x1 + x2 < 2
x1, x2 > 0
If even one _____________constraint, objective function, or both exists, then it is a nonlinear model. This is called a ________________(NLP).
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To use the _______approach, first we graph each constraint to find the _____________(constraint setconstraint set) which is the set of points that simultaneously satisfy _____the constraints.This set represents the ___________decisions.
We want to find the allowable decision that ___________the objective function. To do this, find the “most uphill” _________of the objective function that still touches the constraint set.
The point at which it touches will be an ________ solution to the model.
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x2
x1
2
2There is only one __________constraint and
the solution is not at a corner intersection.
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Noncorner Optima:Noncorner Optima:This graph shows
a ________________ nonlinear inequality
constrained ___________model in
which all constraints are linear and the
constraint set is a standard
LP_____________.
x2
x1
The objective function is __________.
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Nonlinear Optimization: Nonlinear Optimization: A Descriptive Geometric A Descriptive Geometric
IntroductionIntroduction
The following statements hold in either LP or NLP models:
Comparisons between LP and NLP:Comparisons between LP and NLP:
1. Increasing (decreasing) the ____of a < (>) constraint loosens the constraint. This cannot contract and may expand the _____________set.2. Increasing (decreasing) the RHS of a < (>) constraint __________the constraint. This cannot expand and may ____________the constraint set.
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3. ________a constraint cannot hurt and may help the optimal ____________value.4. ___________a constraint cannot help and may hurt the optimal objective value.
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In LP, the ____________on a specified constraint was defined as the rate of the rate of change in OV as the RHS of that change in OV as the RHS of that constraint_________constraint_________, with all other data unchanged.
Lagrange Multiplier:Lagrange Multiplier:
In the NLP context, this rate of change is called the__________________.
In an LP, the shadow price is ___________for a range of values for the RHS parameter of interest. In the NLP context, this property does not generally hold true. Consider the following simple NLP:Max x2
s.t. x < b
x > 0
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In order to ___________x2, make x as large as possible. Thus, the optimal solution is x* = b, and the optimal value of the _______function (OV) is (x*)2 = b2. Thus, the OV is a function of b;OV(b) = b2 .The rate of change of this OV function as b increases is the _________of OV(b), namely 2b. The Lagrange multiplier is not ____________for a range of values of the RHS, b. It varies continuously with b, as might be expected.
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In an LP, it is always true that there cannot be a ______solution that is not also________.
Local versus Global Solutions:Local versus Global Solutions:
This is not usually true for ________NLP models. Such models may have local as well as global solutions. Consider the ___________Max model:x2
x1
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In the previous graph, the point called “Local Max” is termed a local constrained local constrained ___________ ___________ because the value of the objective function at this point is no smaller than at its ____________ feasible points.
The point called “__________” is termed a global constrained maximizerglobal constrained maximizer because the value of the ________function at this point is no smaller than at all other feasible points.
In general for NLPs, additional conditions must be imposed upon the model, called ________and concavity conditions. These conditions must be satisfied to guarantee that a local constrained _____________is also global.
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Nonlinear Optimization: Nonlinear Optimization: A Descriptive Geometric A Descriptive Geometric
IntroductionIntroduction
Many non-linear problems in business and economics are of the following form:
Equality-Constrained NLPs:Equality-Constrained NLPs:
The goal is to maximize or minimize an objective function in n ___________subject to a set of m equality constraints.
Maximize or Minimize f(x1, …, xn)
s.t. g(x1, …, xn) = bi
i = 1, …, m (m<n)
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A manufacturer can make a product on either of two machines.
Example 1.Example 1.
Let x1 denote the _______made on machine 1, x2, the quantity made on machine 2. letax1 + bx1
2 = cost of producing on machine 1
ax2 + bx22 = cost of producing on machine 2
Find the values of x1 and x2 that __________total cost subject to the requirement that total production quantity is to be some given number, say R. The _______________model is:Min ax1 + bx1
2 + ax2 + bx22
s.t. x1 + x2 = R
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The classic _________________model. Let Example 2.Example 2.
Determine the ____________mix that maximizes that person’s utility subject to his/her budget constraint.
p1, p2, and p3 denote given ________prices of three goods
s1, s2, and s3 be given person-specific ____________
B, a specified constant, denotes a person’s available spending____________
x1s1 + x2
s2 + x3s3 denote the person’s
“____” derived from consuming x1 units of good 1, x2 units of good 2 and x3 units of good 3.
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The symbolic model is:
Max x1s1 + x2
s2 + x3s3
s.t. p1x1 + p2 x2 + p3 x3 = B
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Consider the modelExample 3.Example 3.
Max x1 - x2
s.t. -x12 + x2 = 1
x2
x1
1.5
1.0
0.5
0.5 1.0
Solution
EqualityConstraint
Optimal objective contour and constraint
line are tangent at optimal solution.
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Here is the Solved spreadsheet model and formulas:
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Using Solver for NLP ModelsUsing Solver for NLP Models
Solver can be used to enter and ________a model that could contain a nonlinear objective or nonlinear constraint functions or_______.For LP optimization, Solver uses the _________ method to move from corner to corner in the __________region.
For NLP optimization, Solver uses a hill-climbing technique based on a “_______________” procedure, called GRG.
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The following steps describe the GRG procedure:Using the _________values of the decision
variables (specified in Solver’s Changing Changing CellsCells field), the procedure finds a feasible solution. From that initial starting point, a ________is computed that most rapidly improves the OV. ___________ (i.e., changes in values of the decision variables), is made in that direction until a constraint boundary is encountered or until the ______no longer improves.A new direction is __________from that new point, and the process is repeated and continues until no further improvement in any ____________occurs.
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A restaurant’s average daily expense for advertising is $100, all of which is to be allocated to newspaper ads and radio commercials. Let
Example Nonlinear ModelsExample Nonlinear Models
Marketing Expenditures:Marketing Expenditures:
x1 = avg. no. $/day spent on newspaper adsx2 = avg. no. $/day spent on radio ads
Total annual cost of running the advertising dept:Cost = C(x1, x2) = 20,000 – 440x1 – 300x2 +
20x1 + 12x2 + x1x2
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The goal is to find the restaurant’s ________ allocation that will minimize the total annual cost while adhering to the desired ad ___________of $100 per day. The symbolic model is as follows:Min 20,000 – 440x1 – 300x2 + 20x1 + 12x2 +
x1x2s.t. x1 + x2 = 100
x1, x2 > 0
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Here is the Solved Excel spreadsheet:
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Here are the Solver parameters and the Sensitivity Report: This value
indicates that the initial ___of increase in the annual cost of the adv. dept.
would be about $1195
for each additional
budget dollar spent daily on
____________.
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Lagrange MultipliersLagrange Multipliers in NLP have almost the same interpretation as the ____________in LP.
Example Nonlinear ModelsExample Nonlinear Models
Economic Interpretation of Lagrange Economic Interpretation of Lagrange Multipliers and Reduced Gradients:Multipliers and Reduced Gradients:
At_______, the value of the Lagrange multiplier is the ___________rate of change in the optimal value of the objective function as the ith RHS, bi, is increased, with all other data______________.In economic terminology, the ith Lagrange multiplier reflects the _______________of the ith resource.
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The units of measure are:units of the objective function
units of RHS of constraint i
A __________Lagrange multiplier would indicate that increasing the RHS would initially increase the_____.
A __________Lagrange multiplier would indicate that decreasing the RHS would initially ________ the OV.
The Lagrange multiplier can be used to ________ what will happen to the _________value if the RHS is changed.
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Similar to Reduced Cost, the _______________of a variable relates to the upper or lower bound constraints on __________variables.
A ____________Reduced Gradient indicates that increasing the variable will initially decrease the OV.
A ____________Reduced Gradient indicates that increasing the variable will initially increase the OV.
If a decision variable is at its _______bound, the Reduced Gradient should be ___________for the solution to be optimal in a Max model.Otherwise, decreasing the variable would improve the __________function value.
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If a decision variable is at its lower bound, the Reduced gradient should be __________in a Max model.
Otherwise, _____________the variable would improve the objective function value.
If a decision variable is between its upper and lower bounds, the Reduced gradient should be ______for the solution to be_________.
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Consider the following NLP model of a previous LP model. Let
Astro and Cosmo Revisited:Astro and Cosmo Revisited:
Profit = (PA – 210)A + (PC – 230)C
A = daily production of Astro model TV setsPA = selling price of Astros = 314 – 1.9A +.01A2
C = daily production of Cosmo model TV setsPC = selling price of Cosmos = 243 - .14C
If the unit variable cost of an Astro is $210 and the unit variable cost of a Cosmo is $230, then the total profit is
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Max (PA – 210)A + (PC – 230)C
Here is the symbolic model:
PA = .01A2 – 1.9A + 314 (selling price of Astros)
s.t.
PC = -.14C + 243 (selling price of Cosmos)A < 70 (capacity of Astro line)C < 50 (capacity of Cosmo line)A + 2C < 120 (department A labor hours)A + C < 90 (department B labor hours)A, PA, C, PC > 0
Nonlinearconstraints
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Here is the Excel spreadsheet model:
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Here are the Solver parameters :
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Here is the Sensitivity Report:
This constraint is_______. The Lagrange multiplier indicates that _________the OV increases at the rate of about $0.86 per unit of additional labor hours in Dept. A.
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This graph shows that for the Astro and Cosmo NLP model, the optimal solution does not occur at a _______of the feasible region, though it is on the boundary.
0
20
50
90
C
A70 90 120
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Solver may not provide the optimal solution with NLP models. Here are some examples:
Example Nonlinear ModelsExample Nonlinear Models
Optimality in NLPs:Optimality in NLPs:
Gulf Coast Oil blends gasoline from three components:
Gulf Coast Oil Model:Gulf Coast Oil Model:
Domestic blend
Foreign blend (a blending of two sources)Octane Additive
(used only in premium gasoline)
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The Foreign Blend is transported monthly to Gulf Coast Oil in the single 8,000,000 gallon storage compartment of a large tanker.
Component Octane Cost Availability
No. per Gallon (000s Gal/Mth)
Domestic Blend 85 $0.6510,000
Foreign Blend Source 1 93 $0.80
*
Source 2 97 $0.90 *
Premium Additive 900 $30 50
*Because of the way Gulf Coast Oil obtains Source 1 and Source 2, no more than 8,000,000 gallons of Source 1 plus Source 2 may be obtained per month.
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Because the oil purchased from the two sources loses its separate identities when “_______” into the storage compartment of the tanker, the model is called a_______________.The goal is to determine how many gallons of Regular, Midgrade, and Premium gasoline to ________each month, given that it must honor minimum supply contracts of 100 thousand gallons of each type of gasoline.
In addition, each gasoline is subject to a ________octane requirement. The octane number is a weighted average of the octane numbers of its components where the weights are the ________of each component in the blend.
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Minimum Octane No Wholesale Price Per Gallon
Regular 87 $1.18
Midgrade 89 $1.25
Premium 94 $1.40
Let
R = thousand gal. of regularregular gasoline producedM = thousand gal. of midgrademidgrade gas producedP = thousand gallons of premiumpremium gas producedD = thousand gallons of domesticdomestic blend producedA = thousand gallons of premium additiveadditive produced
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RD = thousand gallons of domestic blend in regular gasolineRF = thousand gallons of foreign blend in regular gasoline
MD = thousand gallons of domestic blend in midgrade gasoline
PD = thousand gallons of domestic blend in premium gasolinePF = thousand gallons of foreign blend in premium gasoline
MF = thousand gallons of foreign blend in midgrade gasoline
S1= thous. gal. purchased, foreign source 1S2= thous. gal. purchased, foreign source 2
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OCT = octane number of pooled foreign blend= 93S1 + 97S2
S1 + S2
OCT(S1 + S2) = 93S1 + 97S2In addition to the nonnegativity constraints, the symbolic model is:Max 1.18R + 1.25M + 1.40P - .65D - .8S1 - .9S2
– 30As.t. R = RD + RF (composition of reg. gas)M = MD + MF (composition of
midgrade gas)P = PD + PF + A (composition of premium gas)D = RD + MD + PD (total domestic blend used)S1 + S2 < 8,000
(tanker capacity for foreign sources)
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85RD + OCT*RF > 87R (min octane number for regular
gasoline)
The next 4 constraints are nonlinear:
85MD + OCT*MF > 89M (min octane number for midgrade
gasoline)85PD + OCT*PF + 900A > 94P
(min octane number for premium gasoline)OCT(S1 + S2) = 93S1 + 97S2
(pooling constraint for foreign sources)
D < 10,000 (supply limit for domestic blend)A < 50 (supply limit for premium blend)R, M, and P each > 100 (contract delivery min.)
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Here is an Excel model with example decision values:
Solver will use these values to find an initial starting point.
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Here are the Solver parameters:
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This is the first solution found by Solver:
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The previous Solver ResultsSolver Results message means that Solver has __________its search because the rate of change in the OV was below the Solver _________Solver _________value for 5 iterations (i.e., the rate of improvement in the OV was too low to continue the optimization method).
For NLPs, Solver always starts from a given ______point. Now, run Solver again to force it to begin optimization again to see if it will _______ upon its solution.
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This is the second solution found by Solver:
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Because of the form of the nonlinear constraints, this particular Gulf Coast Oil model is called a ___________modelmodel.
The starting point for the NLP method is very important for a nonconcave model. You may need to try several different _______points. The best starting points are those near the ______ optimal solution to the model.For nonconcave models, there is no ___________ that the Solver solution is the global optimal one.In the Gulf Coast Oil example, after two attempts to _______the model, Solver has converged to a locallocal optimum and not the globalglobal optimum.
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The global optimum for this model is ________ to be the following:
This solution was found by re-optimizing the model dozens of times, each time using a different starting point set of _______cell values.
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Here is the Sensitivity Report for the optimal solution:
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For __________or highly nonlinear models, some of the Solver OptionsSolver Options can be used to try and improve Solver’s GRG’s ____________tactics.
Solver Options SettingsSolver Options Settings
This value is used to ________ Solver’s search when the OV is improving very slowly. If the improvement is < the default value of .00001 for
5________, Solver stops. Setting this value smaller
forces Solver to continue the optimization method even if the rate of change in ____is
small.
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Setting the ________option to QuadraticQuadratic forces Solver to approximate its estimates of the variable equations in its one-dimensional searches by a __________function instead of a linear (tangent) one.
Selecting _______forces Solver to produce a more accurate
approximation by estimating each directional
___________using two adjacent points to each iterative
solution point instead of just one.
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These SearchSearch options determine how
Solver chooses the search ________along
which an improvement in the OV will be sought.
This setting determines how closely the _____
calculations must match the RHS in order for a given constraint to be
____________.If a constraint’s LHS differs from its RHS by an amount less than this setting, then the two are considered equal,
and thus, the constraint is_______.
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The settings shown below are suggested for highly __________or nonconcave NLPs.
If the NLP model involves some integer decision variables, then setting the ____________to 0% will force Solver to continue its search.
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End of Part 1Please continue to Part 2