Outer-approximation (OA) - Revision history

Yda5: /* Biological kinetic model */

2021-12-16T04:12:04Z

Biological kinetic model

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	====Biological kinetic model====		====Biological kinetic model====
	Consider a dynamic kinetic model describing the mechanism of a set of biochemical reactions. The goal is to determine the appropriate values of the model coefficients (e.g., rate constants, initial conditions, etc.), so as to minimize the sum-of-squares of the residuals between the simulated data provided by the model and the experimental observations. The problem solution strategy relies on reformulating the nonlinear dynamic optimization problem as a finite dimensional MINLP by applying a complete discretization using orthogonal collocation on finite elements. This MINLP is next solved using an outer approximation algorithm. <ref>Miró, A., Pozo, C., Guillén-Gosálbez, G., Egea, J. A., & Jiménez, L. (2012). [https://bmcbioinformatics.biomedcentral.com/articles/10.1186/1471-2105-13-90 “Deterministic global optimization algorithm based on outer approximation for the parameter estimation of nonlinear dynamic biological systems”]. BMC bioinformatics, 13(1), 1-12. </ref>		Consider a dynamic kinetic model describing the mechanism of a set of biochemical reactions. The goal is to determine the appropriate values of the model coefficients (e.g., rate constants, initial conditions, etc.), so as to minimize the sum-of-squares of the residuals between the simulated data provided by the model and the experimental observations. The problem solution strategy relies on reformulating the nonlinear dynamic optimization problem as a finite dimensional MINLP by applying a complete discretization using orthogonal collocation on finite elements. This MINLP is next solved using the outer approximation algorithm. <ref>Miró, A., Pozo, C., Guillén-Gosálbez, G., Egea, J. A., & Jiménez, L. (2012). [https://bmcbioinformatics.biomedcentral.com/articles/10.1186/1471-2105-13-90 “Deterministic global optimization algorithm based on outer approximation for the parameter estimation of nonlinear dynamic biological systems”]. BMC bioinformatics, 13(1), 1-12. </ref>

	====Material selection====		====Material selection====

Yda5: /* Conclusion */

2021-12-16T04:08:42Z

Conclusion

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	==Conclusion==		==Conclusion==
	An outer approximation method is presented for solving MINLP problems of a particular class. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the outer approximation algorithm effectively exploits the structure of the problems and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a MILP master problem.<ref name=" Grossman" /> It should be noted that although the outer approximation algorithm relies on convexity assumptions in MINLP, it can also be applied to nonconvex problems.<ref>Viswanathan, J., & Grossmann, I. E. (1990). [https://www.sciencedirect.com/science/article/pii/0098135490870854 “A combined penalty function and outer-approximation method for MINLP optimization”]. Computers & Chemical Engineering, 14(7), 769-782.</ref>		The outer approximation method is presented for solving MINLP problems of a particular class. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the outer approximation algorithm effectively exploits the structure of the problems and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a MILP master problem.<ref name=" Grossman" /> It should be noted that although the outer approximation algorithm relies on convexity assumptions in MINLP, it can also be applied to nonconvex problems.<ref>Viswanathan, J., & Grossmann, I. E. (1990). [https://www.sciencedirect.com/science/article/pii/0098135490870854 “A combined penalty function and outer-approximation method for MINLP optimization”]. Computers & Chemical Engineering, 14(7), 769-782.</ref>
	Many of the real-world problems that can be solved through outer approximation involve process synthesis where binary decisions represent the existence of process units and continuous variables represent operating values. Examples of real-world problems illustrated here also include a dynamic biological kinetic model and material selection for thermal insulation.		Many of the real-world problems that can be solved through outer approximation involve process synthesis where binary decisions represent the existence of process units and continuous variables represent operating values. Examples of real-world problems illustrated here also include a dynamic biological kinetic model and material selection for thermal insulation.

	==References==		==References==

Yda5: /* Large Hadron Collider */

2021-12-16T04:06:01Z

Large Hadron Collider

← Older revision		Revision as of 00:06, 16 December 2021
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	Consider a dynamic kinetic model describing the mechanism of a set of biochemical reactions. The goal is to determine the appropriate values of the model coefficients (e.g., rate constants, initial conditions, etc.), so as to minimize the sum-of-squares of the residuals between the simulated data provided by the model and the experimental observations. The problem solution strategy relies on reformulating the nonlinear dynamic optimization problem as a finite dimensional MINLP by applying a complete discretization using orthogonal collocation on finite elements. This MINLP is next solved using an outer approximation algorithm. <ref>Miró, A., Pozo, C., Guillén-Gosálbez, G., Egea, J. A., & Jiménez, L. (2012). [https://bmcbioinformatics.biomedcentral.com/articles/10.1186/1471-2105-13-90 “Deterministic global optimization algorithm based on outer approximation for the parameter estimation of nonlinear dynamic biological systems”]. BMC bioinformatics, 13(1), 1-12. </ref>		Consider a dynamic kinetic model describing the mechanism of a set of biochemical reactions. The goal is to determine the appropriate values of the model coefficients (e.g., rate constants, initial conditions, etc.), so as to minimize the sum-of-squares of the residuals between the simulated data provided by the model and the experimental observations. The problem solution strategy relies on reformulating the nonlinear dynamic optimization problem as a finite dimensional MINLP by applying a complete discretization using orthogonal collocation on finite elements. This MINLP is next solved using an outer approximation algorithm. <ref>Miró, A., Pozo, C., Guillén-Gosálbez, G., Egea, J. A., & Jiménez, L. (2012). [https://bmcbioinformatics.biomedcentral.com/articles/10.1186/1471-2105-13-90 “Deterministic global optimization algorithm based on outer approximation for the parameter estimation of nonlinear dynamic biological systems”]. BMC bioinformatics, 13(1), 1-12. </ref>

	====~~Large Hadron Collider~~====		====Material selection====
	A thermal insulation system uses a series of heat intercepts and surrounding insulators to minimize the power required to maintain the heat intercepts at certain temperatures. The designer chooses the maximum number of intercepts, the thickness and area of each intercept, and the material of intercept from a discrete set of materials. The choice of material affects the thermal conductivity and total mass of the insulation system. Nonlinear functions in the model are required to accurately model system characteristics such as heat flow between the intercepts, thermal expansion, and stress constraints. Integer variables are used to model the discrete choice of the type of material to use in each layer. This problem can be modeled as MINLP and solved by outer approximation. <ref>Leyffer, S., Linderoth, J., Luedtke, J., Miller, A., & Munson, T. (2009, July).[ https://iopscience.iop.org/article/10.1088/1742-6596/180/1/012014/meta “Applications and algorithms for mixed integer nonlinear programming”]. In Journal of Physics: Conference Series (Vol. 180, No. 1, p. 012014). IOP Publishing. </ref>		A thermal insulation system uses a series of heat intercepts and surrounding insulators to minimize the power required to maintain the heat intercepts at certain temperatures. The designer chooses the maximum number of intercepts, the thickness and area of each intercept, and the material of intercept from a discrete set of materials. The choice of material affects the thermal conductivity and total mass of the insulation system. Nonlinear functions in the model are required to accurately model system characteristics such as heat flow between the intercepts, thermal expansion, and stress constraints. Integer variables are used to model the discrete choice of the type of material to use in each layer. This problem can be modeled as MINLP and solved by outer approximation. <ref>Leyffer, S., Linderoth, J., Luedtke, J., Miller, A., & Munson, T. (2009, July).[ https://iopscience.iop.org/article/10.1088/1742-6596/180/1/012014/meta “Applications and algorithms for mixed integer nonlinear programming”]. In Journal of Physics: Conference Series (Vol. 180, No. 1, p. 012014). IOP Publishing. </ref>

Yda5: /* Conclusion */

2021-12-16T04:03:49Z

Conclusion

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	==Conclusion==		==Conclusion==
	An outer approximation method is presented for solving MINLP problems of a particular class. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the outer approximation algorithm effectively exploits the structure of the problems and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a MILP master problem.<ref name=" Grossman" /> It should be noted that although the outer approximation algorithm relies on convexity assumptions in MINLP, it can also be applied to nonconvex problems.<ref>Viswanathan, J., & Grossmann, I. E. (1990). [https://www.sciencedirect.com/science/article/pii/0098135490870854 “A combined penalty function and outer-approximation method for MINLP optimization”]. Computers & Chemical Engineering, 14(7), 769-782.<ref/>		An outer approximation method is presented for solving MINLP problems of a particular class. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the outer approximation algorithm effectively exploits the structure of the problems and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a MILP master problem.<ref name=" Grossman" /> It should be noted that although the outer approximation algorithm relies on convexity assumptions in MINLP, it can also be applied to nonconvex problems.<ref>Viswanathan, J., & Grossmann, I. E. (1990). [https://www.sciencedirect.com/science/article/pii/0098135490870854 “A combined penalty function and outer-approximation method for MINLP optimization”]. Computers & Chemical Engineering, 14(7), 769-782.</ref>
	Many of the real-world problems that can be solved through outer approximation involve process synthesis where binary decisions represent the existence of process units and continuous variables represent operating values. Examples of real-world problems illustrated here also include a dynamic biological kinetic model and material selection for thermal insulation.		Many of the real-world problems that can be solved through outer approximation involve process synthesis where binary decisions represent the existence of process units and continuous variables represent operating values. Examples of real-world problems illustrated here also include a dynamic biological kinetic model and material selection for thermal insulation.

	==References==		==References==

Yda5: /* Conclusion */

2021-12-16T04:00:03Z

Conclusion

← Older revision		Revision as of 00:00, 16 December 2021
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	==Conclusion==		==Conclusion==
	An outer approximation method is presented for solving MINLP problems of a particular class. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the outer approximation algorithm effectively exploits the structure of the problems and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a MILP master problem.<ref name=" Grossman" />		An outer approximation method is presented for solving MINLP problems of a particular class. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the outer approximation algorithm effectively exploits the structure of the problems and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a MILP master problem.<ref name=" Grossman" /> It should be noted that although the outer approximation algorithm relies on convexity assumptions in MINLP, it can also be applied to nonconvex problems.<ref>Viswanathan, J., & Grossmann, I. E. (1990). [https://www.sciencedirect.com/science/article/pii/0098135490870854 “A combined penalty function and outer-approximation method for MINLP optimization”]. Computers & Chemical Engineering, 14(7), 769-782.<ref/>
	Many of the real-world problems that can be solved through outer approximation involve process synthesis where binary decisions represent the existence of process units and continuous variables represent operating values. Examples of real-world problems illustrated here also include a dynamic biological kinetic model and material selection for thermal insulation.		Many of the real-world problems that can be solved through outer approximation involve process synthesis where binary decisions represent the existence of process units and continuous variables represent operating values. Examples of real-world problems illustrated here also include a dynamic biological kinetic model and material selection for thermal insulation.

	==References==		==References==

Yda5: /* Conclusion */

2021-12-16T03:51:35Z

Conclusion

← Older revision		Revision as of 23:51, 15 December 2021
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	==Conclusion==		==Conclusion==
	An outer approximation method is presented for solving MINLP problems of a particular class~~. It consists in building a mixed-integer linear equivalent to the feasible region of the problem by taking tangents at well specified points~~. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the outer approximation algorithm effectively exploits the structure of the problems and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a MILP master problem.<ref name=" Grossman" />		An outer approximation method is presented for solving MINLP problems of a particular class. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the outer approximation algorithm effectively exploits the structure of the problems and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a MILP master problem.<ref name=" Grossman" />
	Many of the real-world problems that can be solved through outer approximation involve process synthesis where binary decisions represent the existence of process units and continuous variables represent operating values. Examples of real-world problems illustrated here also include a dynamic biological kinetic model and material selection for thermal insulation.		Many of the real-world problems that can be solved through outer approximation involve process synthesis where binary decisions represent the existence of process units and continuous variables represent operating values. Examples of real-world problems illustrated here also include a dynamic biological kinetic model and material selection for thermal insulation.

	==References==		==References==

Yda5: /* Conclusion */

2021-12-16T03:48:33Z

Conclusion

← Older revision		Revision as of 23:48, 15 December 2021
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	==Conclusion==		==Conclusion==
	An outer approximation method is presented for solving MINLP problems of a particular class. It consists in building a mixed-integer linear equivalent to the feasible region of the problem by taking tangents at well specified points. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the outer approximation algorithm effectively exploits the structure of the problems and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a MILP master problem.<ref name=" Grossman" />		An outer approximation method is presented for solving MINLP problems of a particular class. It consists in building a mixed-integer linear equivalent to the feasible region of the problem by taking tangents at well specified points. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the outer approximation algorithm effectively exploits the structure of the problems and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a MILP master problem.<ref name=" Grossman" />
			Many of the real-world problems that can be solved through outer approximation involve process synthesis where binary decisions represent the existence of process units and continuous variables represent operating values. Examples of real-world problems illustrated here also include a dynamic biological kinetic model and material selection for thermal insulation.

	==References==		==References==

Yda5: /* Conclusion */

2021-12-16T03:41:56Z

Conclusion

← Older revision		Revision as of 23:41, 15 December 2021
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	==Conclusion==		==Conclusion==
	An outer approximation method is presented for solving MINLP problems of a particular class. It consists in building a mixed-integer linear equivalent to the feasible region of the problem by taking tangents at well specified points. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the ~~proposed~~ algorithm effectively exploits the structure of the problems and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a MILP master problem.<ref name=" Grossman" />		An outer approximation method is presented for solving MINLP problems of a particular class. It consists in building a mixed-integer linear equivalent to the feasible region of the problem by taking tangents at well specified points. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the outer approximation algorithm effectively exploits the structure of the problems and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a MILP master problem.<ref name=" Grossman" />

	==References==		==References==

Yda5: /* Conclusion */

2021-12-16T03:34:41Z

Conclusion

← Older revision		Revision as of 23:34, 15 December 2021
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	==Conclusion==		==Conclusion==
	~~Outer-Approximation~~ is ~~a well known efficient approach~~ for solving ~~convex mixed-integer nonlinear programs (~~MINLP). It consists in building a mixed-integer linear equivalent to the feasible region of the problem by taking tangents at well specified points. ~~Different algorithms have been devised to build~~ the ~~mixed~~-~~integer linear equivalent including an~~ algorithm ~~proposed by by Duran~~ and and ~~Grossmann in 1986~~.<ref name=" ~~Book~~" /> ~~A step-by-step numerical example was presented to illustrate the OA algorithm. Finally, three MINLP applications that can be solved by the OA method were discussed.~~		An outer approximation method is presented for solving MINLP problems of a particular class. It consists in building a mixed-integer linear equivalent to the feasible region of the problem by taking tangents at well specified points. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the proposed algorithm effectively exploits the structure of the problems and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a MILP master problem.<ref name=" Grossman" />

	==References==		==References==

Yda5: /* Applications */

2021-12-16T03:13:51Z

Applications

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	==Applications==		==Applications==
	The ability to accurately model real-world problems has made MINLP an active research area and there exists a vast number of applications in fields such as engineering, computational chemistry, and finance.<ref name "Hijazi">Hijazi, H., Bonami, P., & Ouorou, A. (2010). [http://www.optimization-online.org/DB_FILE/2012/08/3581.pdf “An outer-inner approximation for separable MINLPs”]. LIF.</ref> The followings are some MINLP applications that can be solved by the OA method:		The ability to accurately model real-world problems has made MINLP an active research area and there exists a vast number of applications in fields such as engineering, computational chemistry, and finance.<ref name "Hijazi">Hijazi, H., Bonami, P., & Ouorou, A. (2010). [http://www.optimization-online.org/DB_FILE/2012/08/3581.pdf “An outer-inner approximation for separable MINLPs”]. LIF.</ref> The followings are some MINLP applications that can be solved by the Outer approximation method:

	====Process Synthesis====		====Process Synthesis====