Branch and bound (BB) for MINLP - Revision history

Pnv4 at 04:58, 16 December 2021

2021-12-16T04:58:34Z

← Older revision		Revision as of 00:58, 16 December 2021
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	== Algorithmic Discussion ==		== Algorithmic Discussion ==
	''Relax'' and ''search'' are the basic performances common to algorithms when finding a solution to MINLP. The difference in algorithms occurs depending on how these performances are executed. In convex MINLPs, the most unaffected relaxation is to ignore the totality constraints <math>y\in Z^p</math>. This results in a nonlinear program (NLP) that is solved to global optimality to produce a lower bound on the optimal solution value <math>z_{MINLP}</math>. When the solution <math>(x^0,y^0)</math> is relaxed to have ''<math>y\in Z^p</math><sup>,</sup>'' then <math>(x^0,y^0)</math> is supposed to be the optimal answer of MINLP; otherwise, there is a <math>j\in \{1...p\}</math> with <math>y_j^0\ \nexists\ \Zeta</math> (Cacchiani & D’Ambrosio, 2017). The search is continuous when it comes to ''branch-and-bound'' whereby it creates two sub-problems; one with the additional constraint <math>y_j \geq [y^0_j]</math> and the other one with additional constraint ''<math>y_j \leq [y^0_j]</math>.'' The same procedure is used to solve these two sub-problems, and it results in search-tree sub-problems that are investigated.		''Relax'' and ''search'' are the basic performances common to algorithms when finding a solution to MINLP. The difference in algorithms occurs depending on how these performances are executed. In convex MINLPs, the most unaffected relaxation is to ignore the totality constraints <math>y\in Z^p</math>. This results in a nonlinear program (NLP) that is solved to global optimality to produce a lower bound on the optimal solution value <math>z_{MINLP}</math>. When the solution <math>(x^0,y^0)</math> is relaxed to have ''<math>y\in Z^p</math><sup>,</sup>'' then <math>(x^0,y^0)</math> is supposed to be the optimal answer of MINLP; otherwise, there is a <math>j\in \{1...p\}</math> with <math>y_j^0\ \nexists\ \Zeta</math> (Cacchiani & D’Ambrosio, 2017). The search is continuous when it comes to ''branch and bound'' whereby it creates two sub-problems; one with the additional constraint <math>y_j \geq [y^0_j]</math> and the other one with additional constraint ''<math>y_j \leq [y^0_j]</math>.'' The same procedure is used to solve these two sub-problems, and it results in search-tree sub-problems that are investigated.

	On the other hand, when <math>f(x, y)</math> or <math>g(x, y)</math> are not convex functions, then quality algorithms for handling NLP will assure that convergence will occur to a local minimum only. Therefore, the simple solution will not be capable of providing a lower bound on <math>z_{MINLP}</math>. In this scenario of nonconvex MINLP, an additional relaxation procedure is needed. Generally, relaxation depends on the decomposition of a nonlinear function into elements and for every element, a "convex envelope" relaxation is developed (Cacchiani & D’Ambrosio, 2017). Branch and bound is again used in relaxation where there is the additional complexity that continuous variables ''x'' can need branching to better the convex relaxation of the initial nonconvex function. While solving this class, a difference occurs in the method in which relaxation is developed.		On the other hand, when <math>f(x, y)</math> or <math>g(x, y)</math> are not convex functions, then quality algorithms for handling NLP will assure that convergence will occur to a local minimum only. Therefore, the simple solution will not be capable of providing a lower bound on <math>z_{MINLP}</math>. In this scenario of nonconvex MINLP, an additional relaxation procedure is needed. Generally, relaxation depends on the decomposition of a nonlinear function into elements and for every element, a "convex envelope" relaxation is developed (Cacchiani & D’Ambrosio, 2017). Branch and bound is again used in relaxation where there is the additional complexity that continuous variables ''x'' can need branching to better the convex relaxation of the initial nonconvex function. While solving this class, a difference occurs in the method in which relaxation is developed.

Pnv4 at 04:53, 16 December 2021

2021-12-16T04:53:31Z

← Older revision		Revision as of 00:53, 16 December 2021
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	== Introduction ==		== Introduction ==
	The arrangements of conventional design problems into programming models make room to their definition and execution of their general option solution. Problems organized as a set of continuous variables with binary integer variables are represented by MINLP models ~~for the~~ continuous variables are restricted to a defined constraints while the binary variables shows if a design option is constructed or not. Branch and bound is a method used to solve Mixed Integer Non-Linear Programming (MINLP) models. There is a difference in the exact steps of the algorithm; however, the initial method created in 2000 by Ioannis Akrotirianakis, Istvan Maros, and Berc Rustem uses a general arrangement of the branch and bound that has iterative nature of the outer estimation (Altherr et al., 2019). The general efficiency of the branch and bound algorithm is improved by adding Gomory mixed-integer cuts as an alternative to branching to reduce the number of nodes and Non-Liner Programming (NLP) problems needed to solve the MINLP.		The arrangements of conventional design problems into programming models make room to their definition and execution of their general option solution. Problems organized as a set of continuous variables with binary integer variables are represented by MINLP models. The continuous variables are restricted to defined constraints while the binary variables shows if a design option is constructed or not. Branch and bound is a method used to solve Mixed Integer Non-Linear Programming (MINLP) models. There is a difference in the exact steps of the algorithm; however, the initial method created in 2000 by Ioannis Akrotirianakis, Istvan Maros, and Berc Rustem uses a general arrangement of the branch and bound that has iterative nature of the outer estimation (Altherr et al., 2019). The general efficiency of the branch and bound algorithm is improved by adding Gomory mixed-integer cuts as an alternative to branching to reduce the number of nodes and Non-Liner Programming (NLP) problems needed to solve the MINLP.
	The algorithm for branch and bound (BB) method was originally proposed by A.H. Land and A.G. Doig in 1960 for discrete programming (Jiyao et al., 2014).		The algorithm for branch and bound (BB) method was originally proposed by A.H. Land and A.G. Doig in 1960 for discrete programming (Jiyao et al., 2014).
	[[File:MINLP Illustration.jpg\|frame\|MINLP Branch-and-Bound]]		[[File:MINLP Illustration.jpg\|frame\|MINLP Branch-and-Bound]]

	== Algorithmic Discussion ==		== Algorithmic Discussion ==
	''Relax'' and ''search'' are the basic performances common to algorithms when finding a solution to MINLP. The difference in algorithms occurs depending on how these performances are executed. In convex MINLPs, the most unaffected relaxation is to ignore the totality constraints <math>y\in Z^p</math>. This results in a nonlinear program (NLP) that is solved to global optimality to produce a lower bound on the optimal solution value <math>z_{MINLP}</math>. When the solution <math>(x^0,y^0)</math> is relaxed to have ''<math>y\in Z^p</math><sup>,</sup>'' then <math>(x^0,y^0)</math> is supposed to be the optimal answer of MINLP; otherwise, there is a <math>j\in \{1...p\}</math> with <math>y_j^0\ \nexists\ \Zeta</math> (Cacchiani & D’Ambrosio, 2017). The search is continuous when it comes to ''branch-and-bound'' whereby it creates two sub-problems; one with the additional constraint <math>y_j \geq [y^0_j]</math> and the other one with additional ~~constraints~~ ''<math>y_j \leq [y^0_j]</math>.'' The same procedure is used to solve these two sub-problems, and it results in search-tree sub-problems that are investigated.		''Relax'' and ''search'' are the basic performances common to algorithms when finding a solution to MINLP. The difference in algorithms occurs depending on how these performances are executed. In convex MINLPs, the most unaffected relaxation is to ignore the totality constraints <math>y\in Z^p</math>. This results in a nonlinear program (NLP) that is solved to global optimality to produce a lower bound on the optimal solution value <math>z_{MINLP}</math>. When the solution <math>(x^0,y^0)</math> is relaxed to have ''<math>y\in Z^p</math><sup>,</sup>'' then <math>(x^0,y^0)</math> is supposed to be the optimal answer of MINLP; otherwise, there is a <math>j\in \{1...p\}</math> with <math>y_j^0\ \nexists\ \Zeta</math> (Cacchiani & D’Ambrosio, 2017). The search is continuous when it comes to ''branch-and-bound'' whereby it creates two sub-problems; one with the additional constraint <math>y_j \geq [y^0_j]</math> and the other one with additional constraint ''<math>y_j \leq [y^0_j]</math>.'' The same procedure is used to solve these two sub-problems, and it results in search-tree sub-problems that are investigated.

	On the other hand, when <math>f(x, y)</math> or <math>g(x, y)</math> is not convex functions, then quality algorithms for handling NLP will assure that convergence will occur to a local minimum only. Therefore, the simple solution will not be capable of providing a lower bound on <math>z_{MINLP}</math>. In this scenario of nonconvex MINLP, an additional relaxation procedure is needed. Generally, relaxation depends on the decomposition of a nonlinear function into elements and for every element, a "convex envelope" relaxation is developed (Cacchiani & D’Ambrosio, 2017). Branch and bound is again used in relaxation where there is the additional complexity that continuous variables ''x'' can need branching to better the convex relaxation of the initial nonconvex function. ~~During~~ solving this class, a difference occurs in the method in which relaxation is developed.		On the other hand, when <math>f(x, y)</math> or <math>g(x, y)</math> are not convex functions, then quality algorithms for handling NLP will assure that convergence will occur to a local minimum only. Therefore, the simple solution will not be capable of providing a lower bound on <math>z_{MINLP}</math>. In this scenario of nonconvex MINLP, an additional relaxation procedure is needed. Generally, relaxation depends on the decomposition of a nonlinear function into elements and for every element, a "convex envelope" relaxation is developed (Cacchiani & D’Ambrosio, 2017). Branch and bound is again used in relaxation where there is the additional complexity that continuous variables ''x'' can need branching to better the convex relaxation of the initial nonconvex function. While solving this class, a difference occurs in the method in which relaxation is developed.

	Let ~~have~~ <math>f(x)</math>, <math>h(x)</math> and <math>g(x)</math> ~~that are~~ convex as well as continuous differentiable. To form a general MINLP problem that will be solved using branch and bound method it will be done as shown below:		Let <math>f(x)</math>, <math>h(x)</math> and <math>g(x)</math> be convex as well as continuous differentiable. To form a general MINLP problem that will be solved using the branch and bound method it will be done as shown below:

	<math>\min\ \Zeta(x,y) = C^Ty + f(x)</math>		<math>\min\ \Zeta(x,y) = C^Ty + f(x)</math>
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	<math>y\in0,1 \quad x\in\Chi</math>		<math>y\in0,1 \quad x\in\Chi</math>

	The branch and bound usually ~~follow~~ the outer approximation for MINLP when incorporating Gomory cuts to reduce the search space for the problem (Jaber et al., 2021). The initial developer presented the algorithm as shown in the various methods below.		The branch and bound method usually follows the outer approximation for MINLP when incorporating Gomory cuts to reduce the search space for the problem (Jaber et al., 2021). The initial developer presented the algorithm as shown in the various methods below.

	=== Initialization: Search Tree Root Value ===		=== Initialization: Search Tree Root Value ===
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	<math> \quad -1\leq x_2 \leq 1 </math>		<math> \quad -1\leq x_2 \leq 1 </math>

	To verify if [0.05, 0.5] is optimal relaxed solution (with continuous variables). This is the root node lower bound for the integer solution objective and beginning point for the ~~brand~~ and bound.		To verify if [0.05, 0.5] is optimal relaxed solution (with continuous variables). This is the root node lower bound for the integer solution objective and beginning point for the branch and bound.

	<math>Obj=4(0.5)^4-4(0.5) (0.5)^4+ (0.5)^2+ (0.5)^2-0.5+1=		<math>Obj=4(0.5)^4-4(0.5) (0.5)^4+ (0.5)^2+ (0.5)^2-0.5+1=
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	== Application ==		== Application ==
	The branch and bound can be used in many programming problems because it converts them to mixed-integer nonlinear programming and solves them using the branch and bound method (Zhang et al., 2020). Besides the mathematical application, it can be used in various engineering and design issues.		The branch and bound method can be used in many programming problems because it converts them to mixed-integer nonlinear programming and solves them using the branch and bound method (Zhang et al., 2020). Besides the mathematical application, it can be used in various engineering and design issues.

	== Mathematical Problem ==		== Mathematical Problem ==
	Mathematically, the Branch and bound method for MINLP is commonly used to solve the travelling salesman problem. In 1991, Padberg and Rinaldi created a solution to large-scale symmetric travelling salesman ~~issues~~ utilizing the branch and cut method (Jaber et al., 2021). MINLP problems that can be solved using branch and bound algorithms are; biological models, scheduling and network design (involving finding location).		Mathematically, the Branch and bound method for MINLP is commonly used to solve the travelling salesman problem. In 1991, Padberg and Rinaldi created a solution to a large-scale symmetric travelling salesman issue utilizing the branch and cut method (Jaber et al., 2021). MINLP problems that can be solved using branch and bound algorithms are; biological models, scheduling and network design (involving finding location).

	== Industrial Application ==		== Industrial Application ==
	''' ''' Designing a thermal insulation layer for the huge hadron collider requires branch and method for MINLP. Series of heat intercepts as well as surrounding insulators ~~are used~~ thermal insulation to reduce the power needed to keep the heat intercepts at specific temperatures (Cacchiani & D’Ambrosio, 2017). These issues come up in ~~the plan of~~ superconducting magnetic energy storage systems as well as ~~have been~~ utilized in large huge collider projects.		''' ''' Designing a thermal insulation layer for the huge hadron collider requires the branch and bound method for MINLP. Series of heat intercepts as well as surrounding insulators use thermal insulation to reduce the power needed to keep the heat intercepts at specific temperatures (Cacchiani & D’Ambrosio, 2017). These issues come up in planning superconducting magnetic energy storage systems as well as being utilized in large huge collider projects. Designers select the maximum number of intercepts; the discrete set of materials is selected depending on the thickness and area of every intercept and the intercepted material. The thermal conductivity as well as total mass of the insulation system depend on the selected material (Cacchiani & D’Ambrosio, 2017). Nonlinear functions on the model are needed for accuracy purposes like stress constraints and heat flow between the intercept and thermal expansion. Integer variables are utilized to model the discrete choice of the type of material to utilize in every layer. When branch and bound for MINLP is used, it identifies better materials compared to using other methods. Branch and bound is an improvement because of the capability to handle a more significant amount of intercepts than other methods (Altherr et al., 2019). Therefore, most designers tend to apply this method to ensure that they have an optimal solution.
	Designers select the maximum number of intercepts; the discrete set of materials is selected depending on the thickness and area of every intercept and the intercepted material. The thermal conductivity as well as total mass of the insulation system depend on the selected material ~~selected~~ (Cacchiani & D’Ambrosio, 2017). Nonlinear functions on the model are needed for accuracy purposes like stress constraints, heat flow between the intercept and thermal expansion. Integer variables are utilized to model the discrete choice of the type of material to utilize in every layer.
	When branch and bound for MINLP is used, it identifies better materials compared to using other methods. Branch and bound ~~have better~~ improvement because of ~~their~~ capability to handle a more significant amount of intercepts than other methods (Altherr et al., 2019). Therefore, most designers tend to apply this method to ensure that they have an optimal solution.

	== Conclusion ==		== Conclusion ==
	In general, ~~Branch~~ and bound method ~~follow~~ a scheme that grows exponentially because variables are added to the initial design problem. When Gomory cuts are added, they allow bounds to tighten by decreasing the number of feasible nodes tested and permitting the solver to converge on the optimal solutions with minimum iterations. However, it takes time to form inequalities to production cuts, which can slow down the real resolution of the MINLP problem. Due to this, it should be noted that the larger the problem, the more significant the cuts become.		In general, the branch and bound method follows a scheme that grows exponentially because variables are added to the initial design problem. When Gomory cuts are added, they allow bounds to tighten by decreasing the number of feasible nodes tested and permitting the solver to converge on the optimal solutions with minimum iterations. However, it takes time to form inequalities to production cuts, which can slow down the real resolution of the MINLP problem. Due to this, it should be noted that the larger the problem, the more significant the cuts become.

	== References ==		== References ==

Pnv4 at 19:16, 15 December 2021

2021-12-15T19:16:03Z

← Older revision		Revision as of 15:16, 15 December 2021
Line 1:		Line 1:
	== Introduction ==		== Introduction ==
	The arrangements of conventional design problems into programming models make room to their definition and execution of their general option solution. Problems organized as a set of continuous variables with binary integer variables are represented by MINLP models for the continuous variables are restricted to a defined constraints while the binary variables shows if a design option is constructed or not. Branch and bound is a method used to solve Mixed Integer Non-Linear Programming (MINLP) models. There is a difference in the exact steps of the algorithm; however, the initial method created in 2000 by Ioannis Akrotirianakis, Istvan Maros, and Berc Rustem uses a general arrangement of the branch and bound that has iterative nature of the outer estimation (Altherr et al., 2019). The general efficiency of the branch and bound algorithm is improved by adding Gomory mixed-integer cuts as an alternative to branching to reduce the number of nodes and Non-Liner Programming (NLP) problems needed to solve the MINLP.		The arrangements of conventional design problems into programming models make room to their definition and execution of their general option solution. Problems organized as a set of continuous variables with binary integer variables are represented by MINLP models for the continuous variables are restricted to a defined constraints while the binary variables shows if a design option is constructed or not. Branch and bound is a method used to solve Mixed Integer Non-Linear Programming (MINLP) models. There is a difference in the exact steps of the algorithm; however, the initial method created in 2000 by Ioannis Akrotirianakis, Istvan Maros, and Berc Rustem uses a general arrangement of the branch and bound that has iterative nature of the outer estimation (Altherr et al., 2019). The general efficiency of the branch and bound algorithm is improved by adding Gomory mixed-integer cuts as an alternative to branching to reduce the number of nodes and Non-Liner Programming (NLP) problems needed to solve the MINLP.
			The algorithm for branch and bound (BB) method was originally proposed by A.H. Land and A.G. Doig in 1960 for discrete programming (Jiyao et al., 2014).
	The algorithm for branch and bound (BB) method was originally proposed by A.H. Land and A.G. Doig in 1960 for discrete programming (Jiyao et al., 2014).		[[File:MINLP Illustration.jpg\|frame\|MINLP Branch-and-Bound]]

	== Algorithmic Discussion ==		== Algorithmic Discussion ==

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