Abstract.It is well known that the duality theory for linear programming (LP) is powerful and elegant and lies behind algorithms such as simplex and interior-point methods. However, the standard Lagrangian for nonlinear programs requires constraint qualifications to avoid duality gaps.Semidefinite linear programming (SDP) is a generalization of LP where the nonnegativity constraints are replaced by a semidefiniteness constraint on the matrix variables. There are many applications, e.g., in systems and control theory and combinatorial optimization. However, the Lagrangian dual for SDP can have a duality gap.We discuss the relationships among various duals and give a unified treatment for strong duality in semidefinite programming. These duals guarantee strong duality, i.e., a zero duality gap and dual attainment. This paper is motivated by the recent paper by Ramana where one of these duals is introduced.

In this expository paper, we review a variety of resource allocation problems in which it is desirable to allocate limited resources equitably among competing activities. Applications for such problems are found in diverse areas, including distribution planning, production planning and scheduling, and emergency services location. Each activity is associated with a performance function, representing, for example, the weighted shortfall of the selected activity level from a specified target. A resource allocation solution is called equitable if no performance function value can be improved without either violating a constraint or degrading an already equal or worse-off (i.e., larger) performance function value that is associated with a different activity. A lexicographic minimax solution determines this equitable solution; that is, it determines the lexicographically smallest vector whose elements, the performance function values, are sorted in nonincreasing order. The problems reviewed include large-scale allocation problems with multiple knapsack resource constraints, multiperiod allocation problems for storable resources, and problems with substitutable resources. The solution of large-scale problems necessitates the design of efficient algorithms that take advantage of special mathematical structures. Indeed, efficient algorithms for many models will be described. We expect that this paper will help practitioners to formulate and solve diverse resource allocation problems, and motivate researchers to explore new models and algorithmic approaches.

We consider a revenue management model for pricing a product line with several customer segments under the assumption that customers' product choices are determined entirely by their reservation prices. We highlight key mathematical properties of the maximum utility model and formulate it as a mixed-integer programming problem, design heuristics and valid cuts. We further present extensions of the models to deal with various practical issues arising in applications. Our computational experiments with real data from the tourism sector as well as with the randomly generated data show the effectiveness of our approach.

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We prove that strict complementarity, primal and dual nondegeneracy of optimal solutions of convex optimization problems in conic form are generic properties. In this paper, we say generic to mean that the set of data possessing the desired property (or properties) has strictly larger Hausdorff dimension than the set of data that does not. Our proof is elementary and it employs an important result due to Larman [7] on the boundary structure of convex bodies.