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In this paper, we introduce the concept of trade-off ratio function, which is closely related to the well-known Geoffrion’s proper efficiency for multi-objective optimization problems, and investigate its boundedness property. For linear multi-objective optimization problems, we show that the trade-off ratio function is bounded on the efficient solution set. For piecewise linear multi-objective optimization problems, we show that all efficient solutions are always properly efficient in Borwein’s sense, and moreover, all efficient solutions are properly efficient in Geoffrion’s sense if and only if a recession condition holds. Finally, we provide an example to illustrate that the trade-off ratio function may be unbounded on the efficient solution set to piecewise linear multi-objective optimization problems, even if the recession condition holds, while it is bounded on the supported efficient solution set.

The development of integrated avionics systems and electromagnetic spectrum technology has attracted widespread attention. It has further increased the performance requirements for modulation signal recognition technology in complex electromagnetic environments. Therefore, this paper proposes a deep joint learning technique, including deep representation and low-dimensionality discrimination, to enhance feature stability and environmental adaptability. Specifically, we design a feature learning network based on AlexNet to extract in-depth features and optimize it through parameter-based transfer learning techniques, promote multi-level representation capabilities of features and reduce the sample size requirements. Moreover, we propose a classification algorithm based on kernel collaborative representation and discriminative projection to enhance the ability of low-dimensionality representation and between-class discrimination, which optimized using the mini-batch random gradient descent method. As shown in the simulation, the overall average recognition success rate of this method aiming at twelve radar signal modulation types reaches 97.58% at SNR of -6dB. The results of simulation and analysis demonstrate the superiority of the proposed model in terms of robustness, timeliness, and adaptability to small samples.

Orthogonal frequency division multiplexing (OFDM) has been widely used as radar, communication, and jamming shared waveform. To improve the power utilization and efficiency of radar, communication, and jamming integrated system, we propose an adaptive OFDM shared waveform design method. Considering the problem of jamming path propagation loss and sensitivity of frequency response error of radar and communication channels, we propose a robust OFDM shared waveform design method. Firstly, the suppressed jamming entropy model, the detection probability (Pd) of the stationary point target model, the data information rate (DIR) model with a given bit error rate (BER) are formulated. With the constraints on total power and BER, the adaptive and the robust OFDM optimization problems are formulated. Then we derive the Karush-Kuhn-Tucker (KKT) conditions of the two optimization problems and an improved greedy algorithm is proposed to solve the optimization problems, and the optimal power and bit allocation schemes for the two problems are obtained. Finally, we present several numerical results to demonstrate the effectiveness of the proposed method.

In order to solve the problems of low utilization of transmit power and low efficiency of OFDM radar communication jamming shared signal waveform, an adaptive shared waveform optimization method is proposed. Firstly, the suppression interference entropy model, signal-to-noise ratio (SNR) and data information rate (SIR) models of shared signals are derived. Then, an adaptive OFDM shared signal waveform optimization model is established with the maximum entropy, signal-to-noise ratio and data information rate as the optimization criteria. Then, the KKT condition of the adaptive waveform is derived, and the optimal adaptive waveform is obtained by convex optimization toolkit Power distribution scheme. The results show that the designed waveform has the best jamming performance of radar communication.

Image space analysis is a new tool for studying scalar and vector constrained extremum problems as well as generalized systems. In the last decades, the introduction of image space analysis has shown that the image space associated with the given problem provides a natural environment for the Lagrange theory of multipliers and that separation arguments turn out to be a fundamental mathematical tool for explaining, developing and improving such a theory. This work, with its 3 parts, aims at contributing to describe the state-of-the-art of image space analysis for constrained optimization and to stress that it allows us to unify and generalize the several topics of optimization. In this 1st part, after a short introduction of such an analysis, necessary and sufficient optimality conditions are treated. Duality and penalization are the contents of the 2nd part. The 3rd part deals with generalized systems, in particular, variational inequalities and Ky Fan inequalities. Some further developments are discussed in all the parts.

In Part I, sufficient and necessary optimality conditions and the image regularity conditions of constrained scalar and vector extremum problems are reviewed for Image Space Analysis. Part II presents the main feature of the duality and penalization of constrained scalar and vector extremum problems by virtue of Image Space Analysis. In the light, as said in Part I and Part II, to describe the state of Image Space Analysis for constrained optimization, and to stress that it allows us to unify and generalize the several topics of Optimization, in this Part III, we continue to give an exhaustive literature review on separation functions, gap functions and error bounds for generalized systems. Part III also throws light on some research gaps and concludes with the scope of future research in this area.

Image space analysis has proved to be instrumental in unifying several theories, apparently disjoint from each other. With reference to constraint qualifications/regularity conditions in optimization, such an analysis has been recently introduced by Moldovan and Pellegrini. Based on this result, the present paper is a preliminary part of a work, which aims at exploiting the image space analysis to establish a general regularity condition for constrained extremum problems. The present part deals with scalar constrained extremum problems in a Euclidean space. The vector case as well as the case of infinite-dimensional image will be the subject of a subsequent part.

Aiming at the typical OFDM shared signal, a method of frequency resource optimization based on the complexity of electromagnetic environment is proposed. Firstly, the mathematical model of multi-dimensional elec-tromagnetic environment occupancy in time domain, spa-tial domain and frequency domain is given. Based on the above model, a method of adaptive allocation of sub-carriers in different frequency bands is given. It has a pioneering significance in the optimization of integrated fre-quency resources of radar communication and jamming.

The main purpose of this paper is to study the duality and penalty method for a constrained nonconvex vector optimization problem. Following along with the image space analysis, a Lagrange-type duality for a constrained nonconvex vector optimization problem is proposed by virtue of the class of vector-valued regular weak separation functions in the image space. Simultaneously, some equivalent characterizations to the zero duality gap property are established including the Lagrange multiplier, the Lagrange saddle point and the regular separation. Moreover, an exact penalization is also obtained by means of a local image regularity condition and a class of particular regular weak separation functions in the image space.

By using the generalized Fermat rule, the Mordukhovich subdifferential for maximum functions, the fuzzy sum rule for Frechet subdifferentials and the sum rule for Mordukhovich subdifferentials, we establish a necessary optimality condition for the local weak sharp efficient solution of a constrained multiobjective optimization problem. Moreover, by employing the approximate projection theorem, and some appropriate convexity and affineness conditions, we also obtain some sufficient optimality conditions respectively for the local and global weak sharp efficient solutions of such a multiobjective optimization problem.