Motivated by its possible applications in mechanics and mechanical engineering, in our previous published work (Pitea and Postolache, 2011), we initiated an optimization theory for the second-order jet bundle. We considered the problem of minimization of vectors of curvilinear functionals (well known as mechanical work), thought as multitime multiobjective variational problems, subject to PDE and/or PDI constraints. Within this framework, we introduced necessary optimality conditions. As natural continuation of these results, the present work introduces a study of sufficient efficiency conditions.

As it is known, most of the optimization problems arising in practice have several objectives which have to be optimized simultaneously. These problems, of considerable interest, include various branches of mathematical sciences, engineering design, portfolio selection, game theory, decision problems in management science, web access problems, query optimization in databases, and so forth. Also, such kind of optimization problems arise in wide areas of research for new technology as well. First of all, we have in mind the material sciences where many times optimal estimation of material parameters is required, either nondestructive determination of faults is needed. Next, chemistry which provides a huge class of constrained optimization problems such as the determination of contamination sources given the flow model and the variance of the source. Last, but not least, games theory in the main study is finding optimal wining strategies. For descriptions of the web access problem, the portfolio selection problem, and capital budgeting problem, see [

In time, several authors have been interested in the study of (vector) ratio programs in connection with generalized convexity. This study is motivated by many practical optimization problems whose objective functions are quotients of two functions. For advances on single-objective ratio programs, see [

Despite of these important advances in optimization, our multitime multiobjective problem—required by practical reasons—had not been studied so far. In the problem of our very recent study [

This work extends, generalizes, and further develops our research in [

Our study is encouraged by its possible application, especially in mechanical engineering, where curvilinear integral objectives are extensively used due to their physical meaning as mechanical work. The objective functions of our type play an essential role in mathematical modeling of certain processes in relation with robotics, tribology, engines, and so forth.

This paper aims to establish some new results on nonlinear optimization on the second order jet bundle. It is organized as follows. In Section

Let

Throughout this work, we use the customary relations between two vectors of the same dimension, [

To simplify the presentation, in our subsequent theory, we shall set

A feasible solution

Consider that

To develop our theory, we have to introduce an appropriate generalized convexity.

Let

The functional

Now, we can establish efficiency sufficient conditions for problem (MFP).

Consider the vectors

the functional

the functional

the functional

one of the integrals of (a)—(c) is strictly

Let us suppose that the point

A total divergence is equal to a total derivative.

After integrating by parts the last two integrals, the left-hand side of the previous inequality can be written as the sum of two integrals. The former has as integrant the scalar product between

Replacing the integrals from hypotheses (b) and (c), of Theorem

Let

the functional

the functional

one of the integrals from (a) or (b) is strictlyquasiinvex at the point

In our work [

Since ratio programming problems with objective function of our type arise from applied areas as decision problems in management, game theory, engineering studies, and design, we will orient our future research to the development of (strong) dual program theory for these problems [