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We focus on a new class of integrally convex functions which we call discrete 2-convex functions. Discrete 2-convexity generalizes known classes of integrally convex functions such as the well-established M-/M $${}^\natural $$ -convex and L-/L $${}^\natural $$ -convex functions by Murota et al., the recently investigated globally/locally discrete midpoint convex functions by Moriguchi, Murota, Tamura, and Tardella, the directed discrete midpoint convex functions by Tamura and Tsurumi, and BS $${}^*$$ -convex and UJ-convex functions by one of the authors. We provide a unifying view of all these functions within the class of integrally convex functions having discrete 2-convexity. We also introduce a new subclass of discrete 2-convex functions, called signed discrete 2-convex functions and we consider signed discrete 2-convex functions with a locally hereditary orientation property. We show that parallelogram inequalities, scalability, and proximity hold for such signed discrete 2-convex functions, which include globally/locally discrete midpoint convex functions and directed discrete midpoint convex functions. Hence, our results extend similar results recently established by Moriguchi, Murota, Tamura, and Tardella and by Tamura and Tsurumi.

A balanced transferable utility game (N, v) has a stable core if its core is externally stable, that is, if each imputation that is not in the core is dominated by some core element. Given two payoff allocations x and y, we say that x outvotes y via some coalition S of a feasible set if x dominates y via S and x allocates at least v(T) to any feasible T that is not contained in S. It turns out that outvoting is transitive and the set M of maximal elements with respect to outvoting coincides with the core if and only if the game has a stable core. By applying the duality theorem of linear programming twice, it is shown that M coincides with the core if and only if a certain nested balancedness condition holds. Thus, it can be checked in finitely many steps whether a balanced game has a stable core. We say that the game has a super-stable core if each payoff vector that allocates less than v(S) to some coalition S is dominated by some core element and prove that core super-stability is equivalent to vital extendability, requiring that each vital coalition is extendable.

Integer programming problems that arise in practice often involve decision variables with one or two sided bounds. In this paper, we consider a generalization of Chvátal-Gomory inequalities obtained by strengthening Chvátal-Gomory inequalities using the bounds on the variables. We prove that the closure of a rational polyhedron obtained after applying the generalized Chvátal-Gomory inequalities is also a rational polyhedron. This generalizes a result of Dunkel and Schulz on 0–1 problems to the case when some of the variables have upper or lower bounds or both while the rest of them are unbounded.

A continuous deformation algorithm is proposed for solving a variational inequality problem on a polytopeK. The algorithm embeds the polytopeK intoK× [0, ∞) and starts by applying a variable dimension algorithm onK× {0} until an approximate solution is found onK× {0}. Then by tracing the path of solutions of a system of equations the algorithm virtually follows a path of approximate solution inK× [0, ∞). When the path inK× [0, ∞) returns to level 0, i.e.,K× {0}, the variable dimension algorithm is again used until a new approximate solution is found onK× {0}. The setK× [0, ∞) is triangulated so that the approximate solution on the path improves the accuracy as the level increases. A contrivance for a practical implementation of the algorithm is proposed and tested for some test problems.

Mixed-integer convex representable (MICP-R) sets are those sets that can be represented exactly through a mixed-integer convex programming formulation. Following up on recent work by Lubin et al. (in: Eisenbrand (ed) Integer Programming and Combinatorial Optimization - 19th International Conference, Springer, Waterloo), (Math. Oper. Res. 47:720-749, 2022) we investigate structural geometric properties of MICP-R sets, which strongly differentiate them from the class of mixed-integer linear representable (MILP-R) sets. First, we provide an example of an MICP-R set which is the countably infinite union of convex sets with countably infinitely many different recession cones. This is in sharp contrast with MILP-R sets which are (countable) unions of polyhedra that share the same recession cone. Second, we provide an example of an MICP-R set which is the countably infinite union of polytopes all of which have different shapes (no pair is combinatorially equivalent, which implies they are not affine transformations of each other). Again, this is in sharp contrast with MILP-R sets which are (countable) unions of polyhedra that are all translations of a finite subset of themselves.

Optimal solutions of Linear Programming problems may become severely infeasible if the nominal data is slightly perturbed. We demonstrate this phenomenon by studying 90 LPs from the well-known NETLIB collection. We then apply the Robust Optimization methodology (Ben-Tal and Nemirovski [1–3]; El Ghaoui et al. [5, 6]) to produce “robust” solutions of the above LPs which are in a sense immuned against uncertainty. Surprisingly, for the NETLIB problems these robust solutions nearly lose nothing in optimality.

In a recent Math. Program. paper, Eckstein and Silva proposed a new error criterion for the approximate solutions of augmented Lagrangian subproblems. Based on a saddle-point formulation of the primal and dual problems, they proved that dual sequences generated by augmented Lagrangians under this error criterion are bounded and that their limit points are dual solutions. In this note, we prove a new result about the convergence of Fejér-monotone sequences in product spaces (which seems to be interesting by itself) and, as a consequence, we obtain the full convergence of the dual sequence generated by augmented Lagrangians under Eckstein and Silva’s criterion.

 We propose and analyze a class of penalty-function-free nonmonotone trust-region methods for nonlinear equality constrained optimization problems. The algorithmic framework yields global convergence without using a merit function and allows nonmonotonicity independently for both, the constraint violation and the value of the Lagrangian function. Similar to the Byrd–Omojokun class of algorithms, each step is composed of a quasi-normal and a tangential step. Both steps are required to satisfy a decrease condition for their respective trust-region subproblems. The proposed mechanism for accepting steps combines nonmonotone decrease conditions on the constraint violation and/or the Lagrangian function, which leads to a flexibility and acceptance behavior comparable to filter-based methods. We establish the global convergence of the method. Furthermore, transition to quadratic local convergence is proved. Numerical tests are presented that confirm the robustness and efficiency of the approach.