A note on Lagrange multipliers in the multiple constraint case

Generalized marginal rate of substitution in multiconstraint consumer’s problems and their reciprocal expenditure problems

“…From a practical perspective, these reciprocal problems would be of some interest if the solution to the utility maximization problem were also a solution to all the reciprocal expenditure minimization problems. Caputo (2001) and Besada and Mirás (2002) provided results in this line that we extend to the general framework of problems P(c) and R(u, c −k ).…”

“…Applying the main theorem of Besada and Mirás (2002) we know that x * = x(c * ) is a minimum of all the equality constraint reciprocal expenditure problems…”

“…More recently, Partovi and Caputo (2006) present a new comparative statics formalism for any differentiable, constraint optimization problem. Weber (1998Weber ( , 2001Caputo (2001) and Besada and Mirás (2002) study the relationships between the utility maximization problem and the, so called, reciprocal expenditure or cost minimization problems by analyzing the equality constraint optimization problem formed by those constraints binding at the optimum. They examine whether an optimal solution of the primal problem is also a solution for all the reciprocal problems and study the reciprocal relations among the corresponding multipliers and their interpretation, as in the single constraint case.…”

“…Nevertheless, a lower optimum value does not necessarily imply lower welfare, as could happen in a problem with dietary requirements to improve consumer's health. Therefore, our aim in this paper is to further explore the multiple constraint optimization results of Besada and Mirás (2002);Caputo (2001);Weber (2001Weber ( , 1998 and Besada and Vázquez (1999) in the context of quasi-concave programming problems.…”

A new proof of the dual optimization problem and its application to the optimal material distribution of SiC/graphene composite

“…The method of Lagrange multipliers as an approach has been extensively practiced in economics (Besada & Mirás, 2002;Beviá & Corchón, 2016;Caputo, 2001;Chatelain, 2000;Haeser & de Melo, 2015;He, 2017;Juhl, 2004;Ponthiere, 2016;Weber, 1998), and its duality principle (Boyd & Vandenberghe, 2004;Magno, et al, 2017) is also widely used in economics. For example, the original optimal problem is to maximize its profit of a factory, then its dual problem is to minimize the cost.…”