Global Optimality Conditions Research Articles

Позиционный принцип минимума: вариационное усиление понятий экстремальности в оптимальном управлении

Existing maximum principles of Pontryagin’s type and related optimality conditions, such as, e.g., the ones derived by F. Clarke, B. Kaskosz and S. Lojasiewicz Jr., and H.J. Sussmann, can be strengthened up to global necessary optimality conditions in the form of so-called feedback minimum principle. This is possible for both classical and non-smooth optimal control problems without terminal constraints. The formulation of the feedback minimum principle (or related extremality conditions) remains within basic constructions of the mentioned maximum principles (the Hamiltonian or Pontryagin function, the adjoint differential equation or inclusion, and its solutions –– co-trajectories). At the same time, the actual maximum condition –– maximization of the Hamiltonian –– takes a variational form: any optimal trajectory of the addressed problem should be optimal for a specific “accessory” problem of dynamic optimization. The latter is stated over all tubes of Krasovskii-Subbotin constructive motions generated by feedback strategies, which are extremal with respect to a certain supersolution of the Hamilton-Jacobi equation. Such a supersolution can be represented explicitely in terms of the co-trajectory of a reference control process and the terminal cost function. In a general version, the feedback minimum principle operates with generalized solutions of the proximal Hamilton-Jacobi inequality for weakly decreasing (𝑢-stable) functions

Global optimality conditions for DC-constrained optimal control problems

Global optimality conditions for DC-constrained optimal control problems

Semi-Supervised Domain Adaptation for Holistic Counting under Label Gap.

This paper proposes a novel approach for semi-supervised domain adaptation for holistic regression tasks, where a DNN predicts a continuous value given an input image x. The current literature generally lacks specific domain adaptation approaches for this task, as most of them mostly focus on classification. In the context of holistic regression, most of the real-world datasets not only exhibit a covariate (or domain) shift, but also a label gap—the target dataset may contain labels not included in the source dataset (and vice versa). We propose an approach tackling both covariate and label gap in a unified training framework. Specifically, a Generative Adversarial Network (GAN) is used to reduce covariate shift, and label gap is mitigated via label normalisation. To avoid overfitting, we propose a stopping criterion that simultaneously takes advantage of the Maximum Mean Discrepancy and the GAN Global Optimality condition. To restore the original label range—that was previously normalised—a handful of annotated images from the target domain are used. Our experimental results, run on 3 different datasets, demonstrate that our approach drastically outperforms the state-of-the-art across the board. Specifically, for the cell counting problem, the mean squared error (MSE) is reduced from 759 to 5.62; in the case of the pedestrian dataset, our approach lowered the MSE from 131 to 1.47. For the last experimental setup, we borrowed a task from plant biology, i.e., counting the number of leaves in a plant, and we ran two series of experiments, showing the MSE is reduced from 2.36 to 0.88 (intra-species), and from 1.48 to 0.6 (inter-species).

Hybrid support vector regression and crow search algorithm for modeling and multiobjective optimization of microalgae-based wastewater treatment

Microalgae-based wastewater treatment (and biomass production) is an environmentally benign and energetically efficient technique as compared to traditional practices. The present study is focused on optimization of the major treatment variables such as temperature, light-dark cycle (LD), and nitrogen (N)-to-phosphate (P) ratio (N/P) for the elimination of N and P from tertiary municipal wastewater utilizing Chlorella kessleri microalgae species. In this regard, a hybrid support vector regression (SVR) technique integrated with the crow search algorithm has been applied as a novel modeling/optimization tool. The SVR models were formulated using the experimental data, which were furnished according to the response surface methodology with Box-Behnken Design. Various statistical indicators, including mean absolute percentage error, Taylor diagram, and fractional bias, confirmed the superior performance of SVR models as compared to the response surface methodology (RSM) and generalized linear model (GLM). Finally, the best SVR model was hybridized with the crow search algorithm for single/multi-objective optimizations to acquire the global optimal treatment conditions for maximum N and P removal efficiencies. The best-operating conditions were found to be 29.3°C, 24/0 h/h of LD, and 6:1 of N/P, with N and P elimination efficiencies of 99.97 and 93.48%, respectively. The optimized values were further confirmed by new experimental data.

Optimal Control Problems Involving Combined Fractional Operators with General Analytic Kernels

Fractional optimal control problems via a wide class of fractional operators with a general analytic kernel are introduced. Necessary optimality conditions of Pontryagin type for the considered problem are obtained after proving a Gronwall type inequality as well as results on continuity and differentiability of perturbed trajectories. Moreover, a Mangasarian type sufficient global optimality condition for the general analytic kernel fractional optimal control problem is proved. An illustrative example is discussed.

Optimal Flow of MVDC Shipboard Microgrids With Hybrid Storage Enhanced With Capacitive and Resistive Droop Controllers

Hybrid storage system composed of battery energy storage systems (BESSs) and supercapacitors is a promising solution to mitigate the high frequency power fluctuations of pulsed power loads (PPLs) in medium voltage DC (MVDC) shipboard power systems. Due to the presence of multiple storage units, a power sharing algorithm needs to be considered within the energy management system (EMS) in the shipboard power system. In this paper, an optimal power flow problem is formulated for MVDC shipboard power systems with hybrid energy storage systems. Battery energy storage systems (BESSs) and conventional generation units are enhanced with virtual resistive droop controllers to share the steady-state power fluctuations. Supercapacitors are enhanced with virtual capacitive droop controllers to share the high frequency fluctuations in the load. The optimal flow accounts for minimizing the operational cost of the generation units in the shipboard power system considering the constraints of network, load balance, voltage profile, and power/energy limits. Second-order cone programming (SOCP) relaxation is used to approximate the nonconvexity of the optimal flow formulation and necessary conditions for global optimality of the solution are discussed. Results confirm the effectiveness of the proposed formulation in load support and managing the energy between storage units.

Development of a Natural Preservative from Chestnut Flowers: Ultrasound-Assisted Extraction Optimization and Functionality Assessment

This study was carried out with the aim of optimizing the ultrasound-assisted extraction (UAE) of phenolic compounds from male chestnut flowers (C. sativa Mill) to develop a bioactive extract with potential to be used as a natural antioxidant preservative ingredient in the food industry. Time (t, 1–39 min), solvent concentration (S, 0–100%), and ultrasonic power (P, 5–500 W) were used as the independent variables for a 5-level experimental circumscribed central composite design (CCCD) coupled with response surface methodology (RSM) to optimize the extraction of phenolic compounds by UAE. Regarding the variables, the three showed a significant effect on the extraction of phenolic compounds. The content of phenolic compounds (including flavonoids and tannins) and the extraction yield (extract weight gravimetrically assessed) were the response criteria for the optimization. Based on the statistically validated predictive polynomial models, it was possible to reach a maximum content of phenolic compounds at the global optimal conditions of 24 ± 3 min, 259 ± 16 W, and 51 ± 7% ethanol. Additionally, pentagalloyl-glucoside and trigalloyl-hexahydroxydiphenoyl-glucoside were the major phenolic compounds identified. The optimized extract was then analyzed for their biological properties. The bioactive potential of the chestnut flower extract obtained under these optimized conditions was evaluated using in vitro assays for antioxidant, anti-inflammatory, and antimicrobial activity, as well as cytotoxicity and hepatotoxicity tests. The results revealed that the enriched extract has antioxidant, antitumoral, and anti-inflammatory activities without toxicity issues. Overall, this study allowed to define the optimal conditions for the extraction of phenolic compounds from chestnuts male flowers by UAE, to obtain an enriched extract with biological properties that could be further used as a natural antioxidant ingredient with applications on functional foods.

On nonconvex optimal control problems

The paper addresses the general optimal control (OC) problem with inequality constraints and a cost functional of Bolza given by d.c. functions with respect to the state in the terminal and integrand parts of the functionals. First, we reduce the original OC problem with inequality constraints to the one without constraints with the help of the Exact Penalization Theory. Further, we show that the auxiliary (penalized) problem also possesses the state-DC-structure. Employing this property, we develop the new Global Optimality Conditions (GOCs) and discuss some its features allowing to construct the new schemes of local and global searchers. Finally we elucidate the relations of the GOCs to the classical OC theory, in particular, to the Pontryagin’s maximum principle.

Necessary and Sufficient Optimality Conditions in DC Semi-infinite Programming

This paper deals with particular families of DC optimization problems involving suprema of convex functions. We show that the specific structure of this type of function allows us to cover a variety of problems in nonconvex programming. Necessary and sufficient optimality conditions for these families of DC optimization problems are established, where some of these structural features are conveniently exploited. More precisely, we derive necessary and sufficient conditions for (global and local) optimality in DC semi-infinite programming and DC cone-constrained optimization, under natural constraint qualifications. Finally, a penalty approach to DC abstract programming problems is developed in the last section.

Some Convexificators-Based Optimality Conditions for Nonsmooth Mathematical Program with Vanishing Constraints

In this paper, by using the notion of convexificator, we introduce the generalized standard Abadie constraint qualification and the generalized MPVC Abadie constraint qualification, and define the generalized stationary conditions for the nonsmooth mathematical program with vanishing constraints (MPVC for short). We show that the generalized strong stationary is the first order necessary optimality condition for nonsmooth MPVC under the generalized standard Abadie constraint qualification. Sufficient conditions for global or local optimality for nonsmooth MPVC are also derived under some generalized convexity assumptions.

Global optimality condition for quadratic optimization problems under data uncertainty

In this paper, we establish a robust version of the S-lemma that we use to characterize robust solutions for classes of homogeneous and non-homogeneous quadratic problems with a quadratic inequality constraint under interval uncertainty and a linear equality constraint. Necessary and sufficient conditions of global optimality of robust solution of these problems are given.

Quadratic problems with two quadratic constraints: convex quadratic relaxation and strong lagrangian duality

In this paper, we study a nonconvex quadratic minimization problem with two quadratic constraints, one of which being convex. We introduce two convex quadratic relaxations (CQRs) and discuss cases, where the problem is equivalent to exactly one of the CQRs. Particularly, we show that the global optimal solution can be recovered from an optimal solution of the CQRs. Through this equivalence, we introduce new conditions under which the problem enjoys strong Lagrangian duality, generalizing the recent condition in the literature. Finally, under the new conditions, we present necessary and sufficient conditions for global optimality of the problem.

Optimality and duality for nonsmooth semi-infinite mathematical program with equilibrium constraints involving generalized invexity of order σ > 0

In this paper, we derive sufficient condition for global optimality for a nonsmooth semi-infinite mathematical program with equilibrium constraints involving generalized invexity of order σ > 0 assumptions. We formulate the Wolfe and Mond–Weir type dual models for the problem using convexificators. We establish weak, strong and strict converse duality theorems to relate the semi-infinite mathematical program with equilibrium constraints and the dual models in the framework of convexificators.

On global optimality conditions for D.C. minimization problems with D.C. constraints

On global optimality conditions for D.C. minimization problems with D.C. constraints

A variational asymmetric phase-field model of quasi-brittle fracture: Energetic solutions and their computation.

We derive the variational formulation of a gradient damage model by applying the energetic formulation of rate-independent processes and obtain a regularized formulation of fracture. The model exhibits different behaviour at traction and compression and has a state-dependent dissipation potential which induces a path-independent work. We will show how such formulation provides the natural framework for setting up a consistent numerical scheme with the underlying variational structure and for the derivation of additional necessary conditions of global optimality in the form of a two-sided energetic inequality. These conditions will form our criteria for making a better choice of the starting guess in the application of the alternating minimization scheme to describe crack propagation as quasistatic evolution of global minimizers of the underlying incremental functional. We will apply the procedure for two- and three-dimensional benchmark problems and we will compare the results with the solution of the weak form of the Euler-Lagrange equations. We will observe that by including the two-sided energetic inequality in our solution method, we describe, for some of the benchmark problems, an equilibrium path when damage starts to manifest, which is different from the one obtained by solving simply the stationariety conditions of the underlying functional.

Short-term oil production global optimization with operational constraints: A comparative study of nonlinear and piecewise linear formulations

The short-term oil production optimization of offshore platforms which use continuous gas lift and consider several operational constraints is a challenging task, specially when decision variables include both continuous and integer ones. The intrinsic non-linearity of the Gas Lift Performance Curves (GLPC) naturally yields to Mixer-Integer Nonlinear Program formulation. Picewise linear approximations of the GLPC are usually applied in order to reduce the complexity of the optimization problem and to ensure an approximate solution, leading to Mixer-Integer Linear Program formulation. However, a comprehensive and comparative study of these two formulations for the oil production optimization problem is scarce in the literature. This paper provides a deep performance evaluation of these two approaches, showing the limitations of both strategies, and proposes a hybrid two-stage optimization strategy to achieve the global optimal operating condition which maximizes the oil production subject to safety, environmental and processing capacity constraints.

Predictive Energy Requirement Models in Bio-fiber Comminution Process

Comminution of bio-fibers is necessary for their effective application in development of reinforced composite and feedstock; however, comminution is an energy intensive process and minimization of specific energy required in the size reduction of bio-fibers is an optimization task in design of various particulate based materials. The objective of this study was to model specific energy requirement in bio-fiber comminution process. The banana and coir bio-fibers used in the study were respectively extracted from post-harvest banana stem and outer shell of a coconut. Characterization of treated bio-fibers was done using Fourier Transform Infrared Spectroscopy and comminution process was carried out in a laboratory-scale knife mill. Electrical energy was recorded using a plug-in power energy meter while Kick’s, Rittinger’s and Bond’s energy laws provide insight into the relationship between particle size reduction and energy consumption during comminution process. The Rosin Rammler-Bennett (RRB) and Gaudin-Schuhmann (GS) models were fitted with MATLAB® R2014a. The Artificial Neural Networks (ANN) and Response Surface Method (RSM) was utilised in predicting specific energy requirements as controlled by fiber treatment conditions, milling speed, initial fiber length and drying temperature. The predictive accuracy of the RSM and ANN models were adjudged adequate in terms of the process independent variables. A global optimal condition was found at 2% NaOH concentration, 29 mm fiber length, 1981 rpm mill speed and 41 °C drying temperature. At these conditions, specific energy was 1.430 kJ/kg for banana fiber and 1.109 kJ/kg for coir fiber having desirability of unity.

The Generalized Trust-Region Sub-Problem with Additional Linear Inequality Constraints—Two Convex Quadratic Relaxations and Strong Duality

In this paper, we study the problem of minimizing a general quadratic function subject to a quadratic inequality constraint with a fixed number of additional linear inequality constraints. Under a regularity condition, we first introduce two convex quadratic relaxations (CQRs), under two different conditions, that are minimizing a linear objective function over two convex quadratic constraints with additional linear inequality constraints. Then, we discuss cases where the CQRs return the optimal solution of the problem, revealing new conditions under which the underlying problem admits strong Lagrangian duality and enjoys exact semidefinite optimization relaxation. Finally, under the given sufficient conditions, we present necessary and sufficient conditions for global optimality of the problem and obtain a form of S-lemma for a system of two quadratic and a fixed number of linear inequalities.

A sufficient conditions for global quadratic optimization

This paper is devoted to global optimality conditions for quadratic optimization problems in a real space of dimension n. More precisely, we are concerned with nonconvex quadratic optimization problems with linear constraints. We present some sufficient conditions of global optimality for such problems subject to linear equality and inequality constraints. We prove that when the set of Karush-Kuhn-Tucker triplets of this problem is convex, then a local minimizer is global

Optimality conditions for efficient solutions of nonconvex constrained multiobjective optimization problems via image space analysis

This paper mainly focuses on optimality conditions for efficient solutions of a nonconvex constrained multiobjective optimization problem via image space analysis. By virtue of the Gerstewitz function and the sum of the components of the objective function, a nonlinear regular weak separation function for efficient solutions, is constructed. In order to illustrate the importance of the sum of the components of the objective function in the nonlinear regular weak separation function for efficient solutions, a nonlinear regular weak separation function for weak efficient solutions, is also given. Moreover, a nonlinear strong separation function for efficient and weak efficient solutions, is introduced. Then, some theorems of the weak and strong alternative and a global necessary and sufficient optimality condition for efficient solutions are derived by means of the nonlinear regular weak and strong separation functions for efficient solutions. A saddle point sufficient optimality condition for efficient solutions in terms of a generalized Lagrangian function associated with the nonlinear regular weak separation function, is established. Finally, under some suitable assumptions, a saddle point necessary optimality condition is obtained, which further yields a Karush/Kuhn–Tucker necessary optimality condition for efficient solutions by the Clarke subdifferential.