Tadeusz Antczak

Multiobjective programming under d-invexity

Abstract In this paper, a generalization of convexity is considered in the case of nonlinear multiobjective programming problem where the functions involved are nondifferentiable. By considering the concept of Pareto optimal solution and substituting d -invexity for convexity, the Fritz John type and Karush–Kuhn–Tucker type necessary optimality conditions and duality in the sense of Mond–Weir and Wolfe for nondifferentiable multiobjective programming are given.

On G-invex multiobjective programming. Part II. Duality

This paper represents the second part of a study concerning the so-called G-multiobjective programming. A new approach to duality in differentiable vector optimization problems is presented. The techniques used are based on the results established in the paper: On G-invex multiobjective programming. Part I. Optimality by T.Antczak. In this work, we use a generalization of convexity, namely G-invexity, to prove new duality results for nonlinear differentiable multiobjective programming problems. For such vector optimization problems, a number of new vector duality problems is introduced. The so-called G-Mond–Weir, G-Wolfe and G-mixed dual vector problems to the primal one are defined. Furthermore, various so-called G-duality theorems are proved between the considered differentiable multiobjective programming problem and its nonconvex vector G-dual problems. Some previous duality results for differentiable multiobjective programming problems turn out to be special cases of the results described in the paper.

A New Approach to Multiobjective Programming with a Modified Objective Function

In this paper, optimality for multiobjective programming problems having invex objective and constraint functions (with respect to the same function η) is considered. An equivalent vector programming problem is constructed by a modification of the objective function. Furthermore, an η-Lagrange function is introduced for a constructed multiobjective problem and modified saddle point results are presented.

Enzymatic lactonization of 15-hydroxypentadecanoic and 16-hydroxyhexadecanoic acids to macrocyclic lactones

Abstract It has been demonstrated that Mucor javanicus L46 and Mucor miehei catalyse the lactonization reaction of 15-hydroxypentadecanoic and 16-hydroxyhexadecanoic acids to appropriate macrocyclic mono- and oligolactones. It has been found that petroleum ether and toluene and their mixture having the polarity log P = 2.96 form an advantageous environment for the reaction. The highest efficiency of synthesis is for toluene at 80°C and for petroleum ether at 70°C. The pH of the essential water layer, as well as the addition of pyridine or DMF, exerts an apparent influence on the reaction rate. The yield of monolactone synthesis is from a few percent to over 30%.

On G-invex multiobjective programming. Part I. Optimality

In this paper, a generalization of convexity, namely G-invexity, is considered in the case of nonlinear multiobjective programming problems where the functions constituting vector optimization problems are differentiable. The modified Karush-Kuhn-Tucker necessary optimality conditions for a certain class of multiobjective programming problems are established. To prove this result, the Kuhn-Tucker constraint qualification and the definition of the Bouligand tangent cone for a set are used. The assumptions on (weak) Pareto optimal solutions are relaxed by means of vector-valued G-invex functions.

Exact penalty functions method for mathematical programming problems involving invex functions

In this paper, some new results on the exact penalty function method are presented. Simple optimality characterizations are given for the differentiable nonconvex optimization problems with both inequality and equality constraints via exact penalty function method. The equivalence between sets of optimal solutions in the original mathematical programming problem and its associated exact penalized optimization problem is established under suitable invexity assumption. Furthermore, the equivalence between a saddle point in the invex mathematical programming problem and an optimal point in its exact penalized optimization problem is also proved.

(p,r)-Invexity in multiobjective programming

Abstract In this paper, a generalization of convexity, namely (p,r)-invexity, is considered in the case of nonlinear multiobjective programming problems where the functions involved are differentiable. The assumptions on Pareto solutions are relaxed by means of (p,r)-invex functions. Also some duality results are obtained for such optimization problems.

Generalized fractional minimax programming with B-(p,r)-invexity

Optimality conditions are proved for a class of generalized fractional minimax programming problems involving B-(p,r)-invexity functions. Subsequently, these optimality conditions are utilized as a basis for constructing various duality models for this type of fractional programming problems and proving appropriate duality theorems.

An η-Approximation Approach for Nonlinear Mathematical Programming Problems Involving Invex Functions

Abstract A new approach to a solution of a nonlinear constrained mathematical programming problem and its Mond–Weir duals is introduced. An η-approximated problem associated with a primal nonlinear programming problem is presented that involves η-approximated functions constituting the primal problem. The equivalence between the original mathematical programming problem and its associated η-approximated optimization problem is established under invexity assumption. Furthermore, η-approximated dual problems in the sense of Mond–Weir are introduced for the obtained η-approximated optimization problem in this method. By the help of η-approximated dual problems some duality results are established for the original mathematical programming problem and its original duals.

Activation of Mucor circinelloides lipase in organic medium

Abstract An intracellular Mucor circinelloides lipase either in the form of mycelium-bound enzyme, or in the homogeneous and soluble form, was subjected to activation experiments. It was found that some compounds, such as pyridine, diethanolamine (DEtA), triethanolamine (TEtA), and cetylpyridinium bromide, either increase or decrease the synthetic lipase activity in organic solvents, dependently on their concentration. Differential spectrophotometry of the homogeneous lipase dissolved in toluene, indicate that the variation in the enzyme activity results from an interaction of these substances with the indole group(s) of the tryptophan residue(s), situated on the surface of this enzyme. Our results prove that M. circinelloides lipase is activated not only in the aqueous milieu, but also in organic systems. The molecular background of this phenomenon seems to be similar to the interfacial activation of lipases in the aqueous system (namely the ‘lid’-helix translocation), thought the reason is different.