Semi-infinite Programming Problem Research Articles

Fractional semi-infinite programming problems: optimality conditions and duality via tangential subdifferentials

Fractional semi-infinite programming problems: optimality conditions and duality via tangential subdifferentials

Lagrange Duality and Saddle-Point Optimality Conditions for Nonsmooth Interval-Valued Multiobjective Semi-Infinite Programming Problems with Vanishing Constraints

This article deals with a class of nonsmooth interval-valued multiobjective semi-infinite programming problems with vanishing constraints (NIMSIPVC). We introduce the VC-Abadie constraint qualification (VC-ACQ) for NIMSIPVC and employ it to establish Karush–Kuhn–Tucker (KKT)-type necessary optimality conditions for the considered problem. Regarding NIMSIPVC, we formulate interval-valued weak vector, as well as interval-valued vector Lagrange-type dual and scalarized Lagrange-type dual problems. Subsequently, we establish the weak, strong, and converse duality results relating to the primal problem, NIMSIPVC, and the corresponding dual problems. Moreover, we introduce the notion of saddle points for the interval-valued vector Lagrangian and scalarized Lagrangian of NIMSIPVC. Furthermore, we derive the saddle-point optimality criteria for NIMSIPVC by establishing the relationships between the solutions of NIMSIPVC and the saddle points of the corresponding Lagrangians of NIMSIPVC, under convexity assumptions. Non-trivial illustrative examples are provided to demonstrate the validity of the established results. The results presented in this paper extend the corresponding results derived in the existing literature from smooth to nonsmooth optimization problems, and we further generalize them for interval-valued multiobjective semi-infinite programming problems with vanishing constraints.

Characterizations of the solution set of nonsmooth semi-infinite programming problems on Hadamard manifolds

This article is concerned with a class of nonsmooth semi-infinite programming problems on Hadamard manifolds (abbreviated as, (NSIP)). We introduce the Guignard constraint qualification (abbreviated as, (GCQ)) for (NSIP). Subsequently, by employing (GCQ), we establish the Karush-Kuhn-Tucker (abbreviated as, KKT) type necessary optimality conditions for (NSIP). Further, we derive that the Lagrangian function associated with a fixed Lagrange multiplier, corresponding to a known solution, remains constant on the solution set of (NSIP) under geodesic pseudoconvexity assumptions. Moreover, we derive certain characterizations of the solution set of the considered problem (NSIP) in the framework of Hadamard manifolds. We provide illustrative examples that highlight the importance of our established results. To the best of our knowledge, characterizations of the solution set of (NSIP) using Clarke subdifferentials on Hadamard manifolds have not been investigated before.

Efficiency conditions and duality for multiobjective semi-infinite programming problems on Hadamard manifolds

Efficiency conditions and duality for multiobjective semi-infinite programming problems on Hadamard manifolds

Lagrange Duality and Saddle Point Optimality Conditions for Multiobjective Semi-Infinite Programming with Vanishing Constraints

The objective of this paper is to investigate multiobjective semi-infinite programming problems with vanishing constraints (in brief, MSIPVC). We firstly propose both the vector Lagrange dual problems and the scalarized Lagrange dual problems for MSIPVC and explore duality relations under convexity assumptions. Then, the saddle point optimality conditions for MSIPVC are considered. Some examples are also given to illuminate the main results.

Approximate optimality conditions and duality results for non-smooth semi-infinite programming problems

This paper concentrates on investigating a semi-infinite optimization problem with cone constraints. Under the attainment of Abadie constraint qualification, an approximate necessary optimality condition by employing Clarke subdifferential for the semi-infinite optimization problem having cone constraints is developed. A new class of functions namely Q-quasiconvex functions is introduced and in the light of the approximate pseudoconvexity and Q-quasiconvexity assumptions, an approximate sufficient optimality condition is investigated using locally Lipschitz functions. Additionally, we formulate the approximate Wolfe's and approximate Mond-Weir dual problems for the non-smooth semi-infinite optimization problem. Subsequently, duality relations in terms of weak, strong and converse results between the semi-infinite optimization model and the aforementioned dual problems are established under the approximate pseudoconvexity and Q-quasiconvexity assumptions. Moreover, to justify the main results of the paper, numerical examples have been shown at suitable places.

Distance-Angle Beamforming for Covert Communications via Frequency Diverse Array: Toward Two-Dimensional Covertness

In this paper, we study the beamforming schemes via the novel frequency diverse array (FDA) on enhancing covert communications performance. We first consider the ideal scenario where the channel state information (CSI) is perfectly known at the transmitter. Then we characterize the key role of minimizing the correlation of the communication and detection channels in boosting the covertness of the system, which also provides a theoretical design principle for FDA-specific carrier frequency scheduling. By exploiting the optimization framework block successive upper bound minimization (BSUM), we propose a frequency scheduling method which leads to a two-phase beamforming scheme to facilitate the covert transmission. Subsequently, we extend the scenario to the more practical one with partial CSI. By leveraging the convex hull, we manage to transform the formulated semi-infinite programming problem to an equivalent semi-definite one which can be solved optimally. In addition, we present a process to construct the optimal beamforming vector. To mitigate the channel correlations in this scenario, we generalize the steps of BSUM and propose an algorithm to schedule the frequencies efficiently. Afterwards, a three-phase robust beamforming scheme is summarized, which boosts the covert rate significantly. Numerical results are provided to demonstrate the superiority of the proposed schemes.

Second-order optimality conditions for interval-valued functions

This work is included in the search of optimality conditions for solutions to the scalar interval optimization problem, both constrained and unconstrained, by means of second-order optimality conditions. As it is known, these conditions allow us to reject some candidates to minima that arise from the first-order conditions. We will define new concepts such as second-order gH-derivative for interval-valued functions, 2-critical points, and 2-KKT-critical points. We obtain and present new types of interval-valued functions, such as 2-pseudoinvex, characterized by the property that all their second-order stationary points are global minima. We extend the optimality criteria to the semi-infinite programming problem and obtain duality theorems. These results represent an improvement in the treatment of optimization problems with interval-valued functions.

The Sufficiency of Solutions for Non-smooth Minimax Fractional Semi-Infinite Programming with (BK,ρ)−Invexity

Minimax fractional semi-infinite programming is an important research direction for semi-infinite programming, and has a wide range of applications, such as military allocation problems, economic theory, cooperative games, and other fields. Convexity theory plays a key role in many aspects of mathematical programming and is the foundation of mathematical programming research. The relevant theories of semi-infinite programming based on different types of convex functions have their own applicable scope and limitations. It is of great value to study semi-infinite programming on the basis of more generalized convex functions and obtain more general results. In this paper, we defined a new type of generalized convex function, based on the concept of the K−directional derivative, that is, uniform (BK,ρ)−invex, strictly uniform (BK,ρ)−invex, uniform (BK,ρ)−pseudoinvex, strictly uniform (BK,ρ)−pseudoinvex, uniform (BK,ρ)−quasiinvex and weakly uniform (BK,ρ)−quasiinvex function. Then, we studied a class of non-smooth minimax fractional semi-infinite programming problems involving this generalized convexity and obtained sufficient optimality conditions.

Optimality conditions and duality for nonsmooth multiobjective semi-infinite programming problems with vanishing constraints on Hadamard manifolds

This article is concerned with nonsmooth multiobjective semi-infinite programming problems with vanishing constraints in the setting of Hadamard manifolds (abbreviated as, (NMSIPVC)). We present the Abadie constraint qualification (abbreviated as, (ACQ)) and a modified version of (ACQ), namely, (NMSIPVC)-tailored Abadie constraint qualification (abbreviated as, (NMSIPVC-ACQ)), for (NMSIPVC). The Karush-Kuhn-Tucker (abbreviated as, KKT) type necessary criteria of optimality for (NMSIPVC) are derived by employing (ACQ) and (NMSIPVC-ACQ). Moreover, sufficient criteria of optimality for (NMSIPVC) are derived under generalized geodesic convexity hypotheses and certain mild restrictions on the index sets. Further, we formulate Wolfe type and Mond-Weir type dual models related to (NMSIPVC) and derive several duality results that relate (NMSIPVC) and the corresponding dual models. Non-trivial numerical examples are incorporated to demonstrate the validity of the results established in this paper. To the best of our knowledge, this is for the first time that constraint qualifications, optimality conditions, and duality results have been investigated for (NMSIPVC) in the setting of Hadamard manifolds.

Second-Order Optimality Conditions and Duality for Multiobjective Semi-Infinite Programming Problems on Hadamard Manifolds

This paper is devoted to the study of multiobjective semi-infinite programming problems on Hadamard manifolds. We consider a class of multiobjective semi-infinite programming problems (abbreviated as MSIP) on Hadamard manifolds. We use the concepts of second-order Karush–Kuhn–Tucker stationary point and second-order Karush–Kuhn–Tucker geodesic pseudoconvexity of the considered problem to derive necessary and sufficient second-order conditions of efficiency, weak efficiency and proper efficiency for MSIP along with certain generalized geodesic convexity assumptions. Moreover, we formulate the second-order Mond–Weir-type dual problem related to MSIP and deduce weak and strong duality theorems relating MSIP and the dual problem. The significance of our results is demonstrated with the help of non-trivial examples. To the best of our knowledge, this is the first time that second-order optimality conditions for MSIP have been studied in Hadamard manifold setting.

Optimality Conditions and Duality for Nonsmooth Multiobjective Semi-infinite Programming Problems on Hadamard Manifolds

Optimality Conditions and Duality for Nonsmooth Multiobjective Semi-infinite Programming Problems on Hadamard Manifolds

A Wasserstein-based distributionally robust neural network for non-intrusive load monitoring

Non-intrusive load monitoring (NILM) is a technique that uses electrical data analysis to disaggregate the total energy consumption of a building or home into the energy consumption of individual appliances. To address the data uncertainty problem in non-intrusive load monitoring, this paper constructs an ambiguity set to improve the robustness of the model based on the distributionally robust optimization (DRO) framework using the Wasserstein metric. Also, for the hard-to-solve semi-infinite programming problem, a novel and computationally efficient upper-layer approximation is used to transform it into an easily solvable regularization problem. Two different data feature extraction methods are used on two open-source datasets, and the experimental results show that the proposed model has good robustness and performs better in identifying devices with large fluctuations. The improvement is about 6% compared to that of the convolutional neural network model without the addition of distributionally robust optimization. The proposed method supports transfer learning and can be added to the neural network in the form of a single-layer net, avoiding unnecessary training times, while ensuring accuracy.

An SDP method for fractional semi-infinite programming problems with SOS-convex polynomials

In this paper, we study a class of fractional semi-infinite polynomial programming problems involving sos-convex polynomial functions. For such a problem, by a conic reformulation proposed in our previous work and the quadratic modules associated with the index set, a hierarchy of semidefinite programming (SDP) relaxations can be constructed and convergent upper bounds of the optimum can be obtained. In this paper, by introducing Lasserre’s measure-based representation of nonnegative polynomials on the index set to the conic reformulation, we present a new SDP relaxation method for the considered problem. This method enables us to compute convergent lower bounds of the optimum and extract approximate minimizers. Moreover, for a set defined by infinitely many sos-convex polynomial inequalities, we obtain a procedure to construct a convergent sequence of outer approximations which have semidefinite representations (SDr). The convergence rate of the lower bounds and outer SDr approximations are also discussed.

On nonsmooth multiobjective semi-infinite programming with switching constraints using tangential subdifferentials

We investigate optimality conditions for a nonsmooth multiobjective semi-infinite programming problem subject to switching constraints. In particular, we employ a surrogate problem and a suitable constraint qualification to state necessary M-stationary conditions in terms of tangential subdifferentials. An example is given at the end to illustrate our main result.

Lower Bound Analysis of Lp+ Induced Norm for LTI Systems

In this paper, we focus on the lower bounds of the Lp (p ∈ [1,∞), p = ∞) induced norms of continuous-time LTI systems where input signals are restricted to be nonnegative. This induced norm, called the Lp+ induced norm, is particularly useful for the stability analysis of nonlinear feedback systems constructed from linear systems and static nonlinearities where the nonlinearities provide only nonnegative signals for the case p = 2. To have deeper understanding on the Lp+ induced norm, we analyze its lower bounds with respect to the standard Lp induced norm in this paper. As the main result, we show that the Lp+ induced norm of an LTI system cannot be smaller than the Lp induced norm scaled by 2(1-p)/p for ∈ [1,∞) (scaled by 2−1 for p = ∞). On the other hand, in the case where p = 2, we further propose a method to compute better (larger) lower bounds for single-input systems via reduction of the lower bound analysis problem into a semi-infinite programming problem. The effectiveness of the lower bound computation method, together with an upper bound computation method proposed in our preceding paper, is illustrated by numerical examples.

New Advances on Fuzzy Linear Programming Problem by Semi-Infinite Programming Approach

As we are faced with more uncertainty problems in the real world, it is necessary to ‎provide ‎models that can provide appropriate solutions for dealing with these issues. In this ‎study, we ‎ proposed a new approach to solving linear programming problem in the fuzzy ‎environment ‎based on solving ‎a related multi-objective model. This kind of problem can be ‎reduced to a ‎fuzzy linear semi-infinite programming problem. In this way, we present a new ‎mixed ‎Multi-Objective Linear Semi-Infinite Programming (MOLSIP) model to solve the ‎main ‎problem, furthermore, as a practical case, we consider a fuzzy Data Envelopment ‎Analysis ‎‎(DEA) model which is a concern to‎ an evaluation of the performance of Decision-‎Making ‎Units (DMUs) in uncertainty environment, The new models show the advantage of ‎our ‎method over the previous ones in terms of certainty. Finally, numerical examples ‎are ‎included to illustrate the suggested solution procedure.‎

Optimality conditions and duality for multiobjective semi-infinite programming problems on Hadamard manifolds using generalized geodesic convexity

This paper deals with multiobjective semi-infinite programming problems on Hadamard manifolds. We establish the sufficient optimality criteria of the considered problem under generalized geodesic convexity assumptions. Moreover, we formulate the Mond-Weir and Wolfe type dual problems and derive the weak, strong and strict converse duality theorems relating the primal and dual problems under generalized geodesic convexity assumptions. Suitable examples have also been given to illustrate the significance of these results. The results presented in this paper extend and generalize the corresponding results in the literature.

Dual Set-Membership State Estimation for Power Distribution Networks Under Event-Triggered Mechanism

This article is concerned with the set-membership state estimation problem for power distribution networks (PDNs) over a resource-constrained communication network under the influence of unknown but bounded (UBB) noises. Firstly, in order to alleviate the pressure of information communication network (ICN) while meeting the state estimation requirements, the event-triggered mechanism is adopted to send data containing more valid information to estimation center, reasonably reducing the signal transmission frequency. Secondly, an event-triggered dual set-membership filter (ET-DSMF) is designed to improve the performance of state estimation. The proposed filter performs a discrete approximation for a semi-infinite programming problem by the random sampling technique, and a compact linearization error bounding ellipsoid is obtained by solving the dual problem of the nonlinear programming. Subsequently, a sufficient condition for the existence of the estimated ellipsoid is derived depending on the mathematical induction method. The key time-varying filter gain matrix and optimal estimated ellipsoid are determined recursively by solving a convex optimization problem, according to the minimum trace criterion. Finally, the effectiveness of the proposed filtering algorithm is demonstrated by performing simulation experiments on the IEEE 13 distribution test system.

Characterizing robust optimal solution sets for nonconvex uncertain semi-infinite programming problems involving tangential subdifferentials

Characterizing robust optimal solution sets for nonconvex uncertain semi-infinite programming problems involving tangential subdifferentials