Comparison of MINLP formulations for global superstructure optimization

Superstructure optimization is a powerful but computationally demanding task that can be used to select the optimal structure among many alternatives within a single optimization. In chemical engineering, such problems naturally arise in process design, where different process alternatives need to be considered simultaneously to minimize a specific objective function (e.g., production costs or global warming impact). Conventionally, superstructure optimization problems are either formulated with the Big-M or the Convex Hull reformulation approach. However, for problems containing nonconvex functions, it is not clear whether these yield the most computationally efficient formulations. We therefore compare the conventional problem formulations with less common ones (using equilibrium constraints, step functions, or multiplications of binary and continuous variables to model disjunctions) using three case studies. First, a minimalist superstructure optimization problem is used to derive conjectures about their computational performance. These conjectures are then further investigated by two more complex literature benchmarks. Our analysis shows that the less common approaches tend to result in a smaller problem size, while keeping relaxations comparably tight—despite the introduction of additional nonconvexities. For the considered case studies, we demonstrate that all reformulation approaches can further benefit from eliminating optimization variables by a reduced-space formulation. For superstructure optimization problems containing nonconvex functions, we therefore encourage to also consider problem formulations that introduce additional nonconvexities but reduce the number of optimization variables.

Optimality conditions for robust weak sharp efficient solutions of nonsmooth uncertain multiobjective optimization problems

In this paper, we investigate an uncertain multiobjective optimization problem involving nonsmooth and nonconvex functions. The notion of a (local/global) robust weak sharp efficient solution is introduced. Then, we establish necessary and sufficient optimality conditions for local and/or the robust weak sharp efficient solutions of the considered problem. These optimality conditions are presented in terms of multipliers and Mordukhovich/limiting subdifferentials of the related functions.

Hermite–Hadamard-Type Inequalities for Generalized Convex Functions via the Caputo-Fabrizio Fractional Integral Operator

Due to applications in almost every area of mathematics, the theory of convex and nonconvex functions becomes a hot area of research for many mathematicians. In the present research, we generalize the Hermite–Hadamard-type inequalities for p , h -convex functions. Moreover, we establish some new inequalities via the Caputo-Fabrizio fractional integral operator for p , h -convex functions. Finally, the applications of our main findings are also given.

Some New Fractional Estimates of Inequalities for LR-p-Convex Interval-Valued Functions by Means of Pseudo Order Relation

It is a familiar fact that interval analysis provides tools to deal with data uncertainty. In general, interval analysis is typically used to deal with the models whose data are composed of inaccuracies that may occur from certain kinds of measurements. In interval analysis, both the inclusion relation ⊆ and pseudo order relation ≤p are two different concepts. In this article, by using pseudo order relation, we introduce the new class of nonconvex functions known as LR-p-convex interval-valued functions (LR-p-convex-IVFs). With the help of this relation, we establish a strong relationship between LR-p-convex-IVFs and Hermite-Hadamard type inequalities (HH-type inequalities) via Katugampola fractional integral operator. Moreover, we have shown that our results include a wide class of new and known inequalities for LR-p-convex-IVFs and their variant forms as special cases. Useful examples that demonstrate the applicability of the theory proposed in this study are given. The concepts and techniques of this paper may be a starting point for further research in this area.

Fractional Integral Inequalities for Exponentially Nonconvex Functions and Their Applications

In this paper, the authors define a new generic class of functions involving a certain modified Fox–Wright function. A useful identity using fractional integrals and this modified Fox–Wright function with two parameters is also found. Applying this as an auxiliary result, we establish some Hermite–Hadamard-type integral inequalities by using the above-mentioned class of functions. Some special cases are derived with relevant details. Moreover, in order to show the efficiency of our main results, an application for error estimation is obtained as well.

A Geometric Integration Approach to Nonsmooth, Nonconvex Optimisation

The optimisation of nonsmooth, nonconvex functions without access to gradients is a particularly challenging problem that is frequently encountered, for example in model parameter optimisation problems. Bilevel optimisation of parameters is a standard setting in areas such as variational regularisation problems and supervised machine learning. We present efficient and robust derivative-free methods called randomised Itoh–Abe methods. These are generalisations of the Itoh–Abe discrete gradient method, a well-known scheme from geometric integration, which has previously only been considered in the smooth setting. We demonstrate that the method and its favourable energy dissipation properties are well defined in the nonsmooth setting. Furthermore, we prove that whenever the objective function is locally Lipschitz continuous, the iterates almost surely converge to a connected set of Clarke stationary points. We present an implementation of the methods, and apply it to various test problems. The numerical results indicate that the randomised Itoh–Abe methods can be superior to state-of-the-art derivative-free optimisation methods in solving nonsmooth problems while still remaining competitive in terms of efficiency.

Some new Jensen, Schur and Hermite-Hadamard inequalities for log convex fuzzy interval-valued functions

<abstract> <p>The inclusion relation and the order relation are two distinct ideas in interval analysis. Convexity and nonconvexity create a significant link with different sorts of inequalities under the inclusion relation. For many classes of convex and nonconvex functions, many works have been devoted to constructing and refining classical inequalities. However, it is generally known that log-convex functions play a significant role in convex theory since they allow us to deduce more precise inequalities than convex functions. Because the idea of log convexity is so important, we used fuzzy order relation $\left(\preceq \right)$ to establish various discrete Jensen and Schur, and Hermite-Hadamard (H-H) integral inequality for log convex fuzzy interval-valued functions (L-convex F-I-V-Fs). Some nontrivial instances are also offered to bolster our findings. Furthermore, we show that our conclusions include as special instances some of the well-known inequalities for L-convex F-I-V-Fs and their variant forms. Furthermore, we show that our conclusions include as special instances some of the well-known inequalities for L-convex F-I-V-Fs and their variant forms. These results and different approaches may open new directions for fuzzy optimization problems, modeling, and interval-valued functions.</p> </abstract>

New integral inequalities for strongly nonconvex functions involving Raina function

In this paper, the authors defined a new general class of functions, the so-called strongly (h1,h2)-nonconvex function involving F??,?(?) (Raina function). Utilizing this, some Hermite-Hadamard type integral inequalities via generalized fractional integral operator are obtained. Some new results as a special cases are given as well.

Global convergence of a modified Broyden family method for nonconvex functions

<p style='text-indent:20px;'>The Broyden family method is one of the most effective methods for solving unconstrained optimization problems. However, the study of the global convergence of the Broyden family method is not sufficient. In this paper, a new Broyden family method is proposed based on the BFGS formula of Yuan and Wei (Comput. Optim. Appl. 47: 237-255, 2010). The following approaches are used in the designed algorithm: (1) a modified Broyden family formula is given, (2) every matrix sequence <inline-formula><tex-math id="M1">\begin{document}$ \{B_k\} $\end{document}</tex-math></inline-formula> generated by the new algorithm possesses positive-definiteness, and (3) the global convergence of the new presented Broyden family algorithm with the Y-W-L inexact line search is obtained for general functions. Numerical performance shows that the modified Broyden family method is competitive with the classical Broyden family method.</p>

Consensus-based optimization on hypersurfaces: Well-posedness and mean-field limit

We introduce a new stochastic differential model for global optimization of nonconvex functions on compact hypersurfaces. The model is inspired by the stochastic Kuramoto–Vicsek system and belongs to the class of Consensus-Based Optimization methods. In fact, particles move on the hypersurface driven by a drift towards an instantaneous consensus point, computed as a convex combination of the particle locations weighted by the cost function according to Laplace’s principle. The consensus point represents an approximation to a global minimizer. The dynamics is further perturbed by a random vector field to favor exploration, whose variance is a function of the distance of the particles to the consensus point. In particular, as soon as the consensus is reached, then the stochastic component vanishes. In this paper, we study the well-posedness of the model and we derive rigorously its mean-field approximation for large particle limit.