When solving a multiobjective optimization problem (MOP) one single optimal solution does not exist. Instead, a set of non-dominated solutions can be obtained, the well-known Pareto optimal front (POF). Decomposition-based methods decompose a MOP into a set of subproblems, each one associated to a weight vector, using a scalarizing function to combine the multiple objectives into a single objective function.
Since the position of the contour lines of the scalarizing function is crucial, a penalty-based weighted Tchebycheff scalarizing function based on the normalized weight vectors is proposed in order to promote convergence to the POF and maintain diversity of solutions on the POF. The influence of the penalty parameter on the opening angle of the contour lines and on the improvement region of a solution is analyzed. Appropriate values for the penalty parameter are obtained.
The proposed algorithm solves each subproblem by the Simulated Annealing to compute a candidate solution. The process is repeated randomly a certain number of times and the final non-dominated solutions are used to approximate the POF.
The tests carried out to optimize a polymer single screw extrusion with two objectives show that the obtained solutions converge well to the POF and have a very good distribution.

In the field of Interval Arithmetic Branch and Bound (B&B) monotonicity detection and its use for dimension reduction and eliminations of partition sets has a long tradition. Recent investigation of extending ideas to simplicial B&B has shown that the relative success depends on the dimension of the simplex, i.e. the dimension of its affine hull. Linear constraints may provide a feasible area which is a polytope in a lower dimension due to equality constraints. We investigate the question how to deal with subsets defined by polytopes. We study how to construct a converging branch and bound, how to define bounds and
how to refine and how to exploit monotonicity when bounds are provided by the interval hull.

Sequential decision-making is crucial in a wide range of industrial problems, including energy expansion planning, power systems operation, and inventory management. We often need to account for uncertainty in model parameters when determining future policies, and multistage stochastic programming provides a powerful framework for doing so. However, solving these models can be challenging, forcing decision makers to rely on heuristics that may produce suboptimal decisions and make it difficult to assess their quality. Two common reasons for the difficulty in solving these problems are discrete decisions such as the installation/expansion of new/existing production units or inventory decisions in discrete production settings, and complex nonlinear (often nonconvex) relationships between decision variables, such as nonlinear cost functions or power flow equations; this leads to mixed-integer nonlinear programming (MINLP) models, which are among the most difficult class of optimization problems to solve. In this work, we address these issues by proposing an exact method for solving multistage stochastic MINLPs with discrete state variables using a tailored column generation (CG) scheme.

Multistage stochastic MINLPs with discrete state variables have an inherently decomposable structure that allows the use of CG-based decomposition. Following a Dantzig-Wolfe reformulation, we apply CG with each pricing subproblem being an MINLP of much smaller size, making it more amenable to global solvers such as Gurobi and BARON. Furthermore, the independent subproblems can be solved in parallel, significantly reducing the solution time. The well-known convergence issues in CG, namely the heading-in and tailing-off effects, are mitigated by column sharing, an information sharing technique among the subproblems that ensures nonanticipativity and allows for the generation of many more feasible columns in each CG iteration. Lastly, we demonstrate the effectiveness of our tailored decomposition scheme through case studies on multistage blending and optimal power flow problem.

A new relaxation technique is developed for the deterministic global optimization of twice-differentiable continuous problems. Instead of using a convex underestimator, the relaxation is based on an underestimator which is edge-concave. The underestimator is constructed by subtracting a positive quadratic expression. We use the linear facets of the vertex polyhedral convex envelope of the edge-concave underestimator to obtain a linear programming (LP)-based relaxation of general nonconvex functions. The method will be presented with theoretical results and will be compared with convexification/ underestimation techniques such as αBB and its variants through test examples.