WE FIRST ESTABLISH SUFFICIENT CONDITIONS ENSURING STRONG DUALITY FOR CONE CONSTRAINED NONCONVEX OPTIMIZATION PROBLEMS UNDER A GENERALIZED SLATER-TYPE CONDITION. SUCH CONDITIONS ALLOW US TO COVER SITUATIONS WHERE RECENT RESULTS CANNOT BE APPLIED. AFTERWARDS, WE PROVIDE A NEW COMPLETE CHARACTERIZATION OF STRONG DUALITY FOR A PROBLEM WITH A SINGLE CONSTRAINT: SHOWING, IN PARTICULAR, THAT STRONG DUALITY STILL HOLDS WITHOUT THE STANDARD SLATER CONDITION. THIS YIELDS LAGRANGE MULTIPLIERS CHARACTERIZATIONS OF GLOBAL OPTIMALITY IN CASE OF (NOT NECESSARILY CONVEX) QUADRATIC HOMOGENEOUS FUNCTIONS AFTER APPLYING A GENERALIZED JOINT-RANGE CONVEXITY RESULT. FURTHERMORE, A RESULT WHICH REDUCES A CONSTRAINED MINIMIZATION PROBLEM INTO ONE WITH A SINGLE CONSTRAINT UNDER GENERALIZED CONVEXITY ASSUMPTIONS, IS ALSO PRESENTED.

GIVEN A NONCONVEX SET-VALUED MAPPINGS F : RN ? R, A NOTION OF CONJUGATE F? IS INTRODUCED WITH THE GOAL THAT (F?)? = F. THIS IS GIVEN BY USING THE USUAL (BILINEAR) DUALITY PAIRING. SEVERAL EXAMPLES SHOWING ITS GEOMETRIC INTERPRETATION ARE PRESENTED, AS WELL AS A NOTION OF SUBDIFFERENTIAL FOR SUCH SET-VALUED MAPS IS ALSO OUTLINED.

SOME PRODUCTION MODELS IN FINANCE REQUIRE INFINITE-DIMENSIONAL COMMODITY SPACES, WHERE EFFICIENCY IS DEFINED IN TERMS OF AN ORDERING CONE HAVING POSSIBLY EMPTY INTERIOR. SINCE WEAK EFFICIENCY IS MORE TRACTABLE THAN EFFICIENCY FROM A MATHEMATICAL POINT OF VIEW, THIS PAPER CHARACTERIZES THE EQUALITY BETWEEN EFFICIENCY AND WEAK EFFICIENCY IN INFINITE-DIMENSIONAL SPACES WITHOUT FURTHER ASSUMPTIONS, LIKE CLOSEDNESS OR FREE DISPOSABILITY. THIS IS OBTAINED AS AN APPLICATION OF OUR MAIN RESULT THAT CHARACTERIZES THE SOLUTIONS TO A UNIFIED VECTOR OPTIMIZATION PROBLEM IN TERMS OF ITS SCALARIZATION. STANDARD MODELS AS EFFICIENCY, WEAK EFFICIENCY (DEFINED IN TERMS OF QUASI-RELATIVE INTERIOR), WEAK STRICT EFFICIENCY, STRICT EFFICIENCY, OR STRONG SOLUTIONS ARE CAREFULLY DESCRIBED. IN ADDITION, WE EXHIBIT TWO PARTICULAR INSTANCES AND COMPUTE THE EFFICIENT AND WEAK EFFICIENT SOLUTION SET IN LEBESGUE SPACES.

SOME PRODUCTION MODELS IN FINANCE REQUIRE INFINITE-DIMENSIONAL COMMODITY SPACES, WHERE EFFICIENCY IS DEFINED IN TERMS OF AN ORDERING CONE HAVING POSSIBLY EMPTY INTERIOR. SINCE WEAK EFFICIENCY IS MORE TRACTABLE THAN EFFICIENCY FROM A MATHEMATICAL POINT OF VIEW, THIS PAPER CHARACTERIZES THE EQUALITY BETWEEN EFFICIENCY AND WEAK EFFICIENCY IN INFINITE-DIMENSIONAL SPACES WITHOUT FURTHER ASSUMPTIONS, LIKE CLOSEDNESS OR FREE DISPOSABILITY. THIS IS OBTAINED AS AN APPLICATION OF OUR MAIN RESULT THAT CHARACTERIZES THE SOLUTIONS TO A UNIFIED VECTOR OPTIMIZATION PROBLEM IN TERMS OF ITS SCALARIZATION. STANDARD MODELS AS EFFICIENCY, WEAK EFFICIENCY (DEFINED IN TERMS OF QUASI-RELATIVE INTERIOR), WEAK STRICT EFFICIENCY, STRICT EFFICIENCY, OR STRONG SOLUTIONS ARE CAREFULLY DESCRIBED. IN ADDITION, WE EXHIBIT TWO PARTICULAR INSTANCES AND COMPUTE THE EFFICIENT AND WEAK EFFICIENT SOLUTION SET IN LEBESGUE SPACES.

ALTERNATIVE THEOREMS HAVE PROVED TO BE IMPORTANT IN DERIVING KEY RESULTS IN OPTIMIZATION THEORY LIKE THE EXISTENCE OF LAGRANGE MULTIPLIERS, DUALITY RESULTS, SCALARIZATION OF VECTOR FUNCTIONS, ETC. SINCE THE PIONEERING RESULT DUE TO JULIUS FARKAS IN 1902 CONCERNING HIS ALTERNATIVE LEMMA WHICH IS WELL KNOWN IN LINEAR PROGRAMMING, OR EVEN THE ELDER ALTERNATIVE RESULT ESTABLISHED BY PAUL GORDAN IN 1873, MANY MATHEMATICIANS HAVE MADE A LOT OF EFFORT TO GENERALIZE BOTH RESULTS IN A NONLINEAR SETTING. TO THESE AUTHOR?S KNOWLEDGE THE FIRST GORDAN TYPE RESULT FOR CONVEX FUNCTIONS IS DUE TO FAN ET AL. [13] AND WAS ESTABLISHED IN 1957. SUCH A RESULT SAYS THE FOLLOWING:

THIS PAPER DEALS WITH MAXIMIZATION AND MINIMIZATION OF QUASICONVEX FUNCTIONS IN A FINITE DIMENSIONAL SETTING. FIRSTLY, SOME EXISTENCE RESULTS ON CLOSED CONVEX SETS, POSSIBLY CONTAINING LINES, ARE PRESENTED. THIS IS GIVEN VIA A CAREFUL STUDY OF REDUCTION TO THE BOUNDARY AND/OR EXTREMALITY OF THE FEASIBLE SET. NECESSARY OR SUFFICIENT OPTIMALITY CONDITIONS ARE DERIVED IN TERMS OF RADIAL EPIDERIVATIVES. THEN, THE PROBLEM OF MINIMIZING QUASICONVEX FUNCTIONS ARE ANALYZED VIA ASYMPTOTIC ANALYSIS. FINALLY, SOME ATTEMPTS TO DEFINE ASYMPTOTIC FUNCTIONS UNDER QUASICONVEXITY ARE ALSO OUTLINED. SEVERAL EXAMPLES ILLUSTRATING THE APPLICABILITY OF OUR RESULTS ARE SHOWN.

PRIMAL OR DUAL STRONG-DUALITY (OR MIN-SUP, INF-MAX DUALITY) IN NONCONVEX OPTIMIZATION IS REVISITED IN VIEW OF RECENT LITERATURE ON THE SUBJECT, ESTABLISHING, IN PARTICULAR, NEW CHARACTERIZATIONS FOR THE SECOND CASE. THIS GIVES RISE TO A NEW CLASS OF QUASICONVEX PROBLEMS HAVING ZERO DUALITY GAP OR CLOSEDNESS OF IMAGES OF VECTOR MAPPINGS ASSOCIATED TO THOSE PROBLEMS. SUCH CONDITIONS ARE DESCRIBED FOR THE CLASSES OF LINEAR FRACTIONAL FUNCTIONS AND THAT OF QUADRATIC ONES. IN ADDITION, SOME APPLICATIONS TO NONCONVEX QUADRATIC OPTIMIZATION PROBLEMS UNDER A SINGLE INEQUALITY OR EQUALITY CONSTRAINT, ARE PRESENTED, PROVIDING NEW RESULTS FOR THE FULFILLMENT OF ZERO DUALITY GAP OR DUAL STRONG-DUALITY.