Scalar Optimization Research Articles

Decomposition-Based-Sorting and Angle-Based-Selection for Evolutionary Multiobjective and Many-Objective Optimization.

Multiobjective evolutionary algorithm based on decomposition (MOEA/D) decomposes a multiobjective optimization problem (MOP) into a number of scalar optimization subproblems and then solves them in parallel. In many MOEA/D variants, each subproblem is associated with one and only one solution. An underlying assumption is that each subproblem has a different Pareto-optimal solution, which may not be held, for irregular Pareto fronts (PFs), e.g., disconnected and degenerate ones. In this paper, we propose a new variant of MOEA/D with sorting-and-selection (MOEA/D-SAS). Different from other selection schemes, the balance between convergence and diversity is achieved by two distinctive components, decomposition-based-sorting (DBS) and angle-based-selection (ABS). DBS only sorts L closest solutions to each subproblem to control the convergence and reduce the computational cost. The parameter L has been made adaptive based on the evolutionary process. ABS takes use of angle information between solutions in the objective space to maintain a more fine-grained diversity. In MOEA/D-SAS, different solutions can be associated with the same subproblems; and some subproblems are allowed to have no associated solution, more flexible to MOPs or many-objective optimization problems (MaOPs) with different shapes of PFs. Comprehensive experimental studies have shown that MOEA/D-SAS outperforms other approaches; and is especially effective on MOPs or MaOPs with irregular PFs. Moreover, the computational efficiency of DBS and the effects of ABS in MOEA/D-SAS are also investigated and discussed in detail.

Self-adaptive multi-objective evolutionary algorithm based on decomposition for large-scale problems: A case study on reservoir flood control operation

Large-scale multi-objective optimization problems (LS-MOP) are complex problems with a large number of decision variables. Due to its high-dimensional decision space, LS-MOP poses a significant challenge to multi-objective optimization methods including multi-objective evolutionary algorithms (MOEAs). Following the algorithmic framework of multi-objective evolutionary algorithm based on decomposition (MOEA/D), an enhanced algorithm with adaptive neighborhood size and genetic operator selection, named self-adaptive MOEA/D (SaMOEA/D), is developed for solving LS-MOP in this work. Learning from the search history, each scalar optimization subproblem in SaMOEA/D varies its neighborhood size and selects a genetic operator adaptively. The former determines the size of the search scope, while the latter determines the search behavior and as a result the newly generated solution. Experimental results on 20 LS-MOP benchmarks have demonstrated that SaMOEA/D outperforms or performs similarly to the other four state-of-the-art MOEAs. The effectiveness of the self-adaptive strategies has also been experimentally verified. Furthermore, SaMOEA/D and the comparing algorithms are then applied to solve a challenging real-world problem, the multi-objective reservoir flood control operation problem. Optimization results illustrate the superiority of SaMOEA/D.

A Multi-Objective Best Compromise Decision Model for Real-Time Flood Mitigation Operations of Multi-Reservoir System

To address the decision-making problem for real-time multi-objective flood operations in multi-reservoir system, this paper develops a multi-objective best compromise decision model (MoBCDM). Utility function is used to quantitatively express the preference of decision maker, and also fuzzy analytic hierarchy process (FAHP) and segmentation and averaging (Seg/Ave) are adopted together with the preferences of decision participants (hydrologist and reservoir manager) to convert the problem into a scalar optimization. The differential evolution (DE) algorithm is implemented for obtaining the best compromise solution. The multi-objective flood operation problem in Shiguan River Basin (in China), which contains two reservoirs and three flood control points, is used as a case study. The analyses are performed to compare four historical flood operations scenarios, this model and current operating rules. The results of the analyses show that the MoBCDM outperforms all operational scenarios in terms of peak flow reduction at three flood downstream control points. In addition, the MoBCDM execution is very efficient in real-time implementation, and also weighting coefficients for the use by the MoBCDM can get high resolution calculated by FAHP.

Energy-Efficient 5G Outdoor-to-Indoor Communication: SUDAS Over Licensed and Unlicensed Spectrum

In this paper, we study the design of the user selection, the time allocation to uplink and downlink, and the transceiver processing matrix for uplink and downlink multicarrier transmission employing a shared user equipment (UE)-side distributed antenna system (SUDAS). The proposed SUDAS simultaneously utilizes licensed frequency bands and unlicensed frequency bands with large available bandwidths (e.g. the millimeter wave bands) to enable a spatial multiplexing gain for single-antenna UEs to improve the energy efficiency and throughput of 5th generation (5G) outdoor-to-indoor communication. The resource allocation algorithm design is formulated as a nonconvex optimization problem for the maximization of the end-to-end system energy efficiency (bits/Joule). The nonconvex matrix optimization problem is converted to an equivalent nonconvex scalar optimization problem for multiple parallel channels, which is solved by an asymptotically globally optimal iterative algorithm. Besides, we propose a suboptimal algorithm, which finds a locally optimal solution of the nonconvex optimization problem. Simulation results illustrate that the proposed resource allocation algorithms for SUDAS achieve a significant performance gain in terms of system energy efficiency and spectral efficiency compared to conventional baseline systems by offering multiple parallel data streams for single-antenna UEs.

Robust optimization revisited via robust vector Farkas lemmas

This paper provides characterizations of the weakly minimal elements of vector optimization problems and the global minima of scalar optimization problems posed on locally convex spaces whose objective functions are deterministic while the uncertain constraints are treated under the robust (or risk-averse) approach, i.e. requiring the feasibility of the decisions to be taken for any possible scenario. To get these optimality conditions we provide Farkas-type results characterizing the inclusion of the robust feasible set into the solution set of some system involving the objective function and possibly uncertain parameters. In the particular case of scalar convex optimization problems, we characterize the optimality conditions in terms of the convexity and closedness of an associated set regarding a suitable point.

Throughput optimisation for energy harvesting transmitter with partial instantaneous channel state information and finite‐alphabet inputs

Wireless communication systems with energy harvester are attracting much attention due to their ability to improve the system operation time. Extensive research has been carried out on how to maximise the sum of mutual information over multiple time slots (i.e. the throughput). However, most research focuses on the energy harvesting transmitter with Gaussian inputs and single antenna. This study considers the throughput maximisation problem for an energy harvesting transmitter with causal energy constraint over multiple-input multiple-output (MIMO) channels. Different from existing works, the authors consider the MIMO system with finite-alphabet inputs and partial instantaneous channel state information (CSI). Specifically, the transmitter knows the statistical CSI of the entire channel frame as well as the instantaneous CSI of current time slot. The precoder design in this scenario is an intractable optimisation problem with respect to multiple precoding matrices. The authors’ analysis shows that this difficult precoding problem can be equivalently transformed into a set of scalar optimisation subproblems with respect to the transmit power. To solve these subproblems, an efficient algorithm based on the dynamic programming is proposed. The authors analyse the performance of the proposed algorithm, and simulation results validate its effectiveness.

CSI Impaired Precoding Optimization for Energy-Efficient MIMO Communications Under Total Power Constraint

This letter considers robust precoding for improving the energy efficiency (EE) in a multiple-input multiple-output (MIMO) channel when only imperfect or partial channel state information (CSI) is available at the transmitter. From the perspective of worst case robustness, we aim to maximize the worst case EE subject to additive or multiplicative bounded CSI uncertainties and under total power constraint. Theoretically, we prove that the robust energy-efficient precoding optimization leads to a channel-diagonalizing structure. Accordingly, the complicated matrix-valued problem is equivalently transformed into a simpler scalar optimization problem, which can be solved efficiently with a water-filling interpretation for the optimal power allocation. The obtained worst case EE was finally verified and evaluated via numerical simulations.

Firefly Algorithm for Finding Optimal Shapes of Electromagnetic Devices

Many real-world optimization problems have to be treated as multi-objective optimization problems. An approach, well established in recent years, is to find Pareto optimal configurations of the trial variables by detecting nondominated solutions with the help of a suitable vector optimization method. Alternatively, relying on scalar optimization methods (both stochastic or deterministic), a suitable objective function taking all objectives into account simultaneously has to be defined. Depending on the number of trial variables, a scalar objective function of that type will exhibit a considerable number of feasible local solutions besides the global one. Therefore, a useful scalar optimization strategy should be able to end up (with a high probability) in the best of all possible solutions in the given search space and additionally detect as many local solutions as possible. Some population-based stochastic methods are implicitly suited for that task; others can be enhanced to fulfill these requirements. Higher order evolution strategies have successfully been tuned for that kind of problem by introducing cluster sensitive recombination [niching higher order evolution strategy (NES)]. The firefly algorithm (FFA) mimics the behavior of fireflies, which use a kind of flashing light to communicate with other members of their species. Since the intensity of the light of a single firefly diminishes with increasing distance, the FFA is implicitly able to detect local solutions on its way to the best solution for a given scalar objective function. The FFA will be applied to the well-known Rastrigin test function and to a shielding/shunting electromagnetic problem with two and three objectives, respectively, and its results will be compared with the ones obtained with an NES.

A scalar optimization approach for averaged Hausdorff approximations of the Pareto front

This article presents a novel method to compute averaged Hausdorff () approximations of the Pareto fronts of multi-objective optimization problems. The underlying idea is to utilize directly the scalar optimization problem that is induced by the performance indicator. This method can be viewed as a certain set based scalarization approach and can be addressed both by mathematical programming techniques and evolutionary algorithms (EAs). In this work, the focus is on the latter where a first single objective EA for such approximations is proposed. Finally, the strength of the novel approach is demonstrated on some bi-objective benchmark problems with different shapes of the Pareto front.

Global efficiency for multiobjective bilevel programming problems under generalized invexity

In this paper, we consider a nonlinear optimistic bilevel programming problem \(({\textit{NMBP}})\) where the upper-level is a vector optimization problem and the lower-level is a scalar optimization problem. By using the Karush–Kuhn–Tucker conditions associated to the lower-level problem, we reformulate the bilevel programming problem into a nonlinear multiobjective single-level programming problem with equality and inequality constraints \(({\textit{MP}})\). Similarly to Dempe and Dutta (2012), we establish relationships between the problems \(({\textit{NMBP}})\) and \(({\textit{MP}})\). We prove that under appropriate constraint qualification and convexity assumptions, global (weakly or properly) efficient solutions of \(({\textit{MP}})\) correspond to global (weakly or properly) efficient solutions of \(({\textit{NMBP}})\). We establish Fritz John type necessary efficiency conditions for \(({\textit{NMBP}})\) without using any constraint qualification. Furthermore, we obtain (Fritz John type) sufficient efficiency conditions for a feasible point of \(({\textit{MP}})\) corresponds to a (weakly or properly) efficient solution for the bilevel problem \(({\textit{NMBP}})\) under various forms of generalized invexity and infineness. Moreover, a linear multiobjective bilevel programming problem is studied and sufficient efficiency conditions are derived. To illustrate the obtained results some examples are given.

Secrecy Capacity Region Maximization in Gaussian MISO Channels with Integrated Services

This letter considers a two-receiver multiple-input single-output Gaussian broadcast channel model with integrated services. Specifically, two sorts of service messages are combined and served simultaneously: one multicast message intended for both receivers and one confidential message intended for only one receiver. The confidential message is kept perfectly secure from the unauthorized receiver. Our goal is to jointly design the input covariances for the multicast message and confidential message, such that the secrecy capacity region is maximized. This secrecy capacity region maximization (SCRM) problem is a nonconvex vector maximization problem. To deal with this issue, we reformulate the SCRM problem into a provably equivalent scalar optimization problem and propose a searching method to find its overall Pareto optimal points. Further, for implementation efficiency, transmit beamforming is proved to be Pareto optimal. However, since the two service messages are coupled in our optimization problem, it is difficult to deduce closed-form expressions of the Pareto optimal beamformers. A suboptimal transmit design is accordingly proposed to analytically obtain beamformers for both service messages. Numerical results illustrate that the performance gap between the Pareto optimal design and our proposal is negligible.

A Survey of Multiobjective Evolutionary Algorithms based on Decomposition

Decomposition is a well-known strategy in traditional multiobjective optimization. However, the decomposition strategy was not widely employed in evolutionary multiobjective optimization until Zhang and Li proposed multiobjective evolutionary algorithm based on decomposition (MOEA/D) in 2007. MOEA/D proposed by Zhang and Li decomposes a multiobjective optimization problem into a number of scalar optimization subproblems and optimizes them in a collaborative manner using an evolutionary algorithm (EA). Each subproblem is optimized by utilizing the information mainly from its several neighboring subproblems. Since the proposition of MOEA/D in 2007, decomposition-based MOEAs have attracted significant attention from the researchers. Investigations have been undertaken in several directions, including development of novel weight vector generation methods, use of new decomposition approaches, efficient allocation of computational resources, modifications in the reproduction operation, mating selection and replacement mechanism, hybridizing decomposition- and dominance-based approaches, etc. Furthermore, several attempts have been made at extending the decomposition-based framework to constrained multiobjective optimization, many-objective optimization, and incorporate the preference of decision makers. Additionally, there have been many attempts at application of decomposition-based MOEAs to solve complex real-world optimization problems. This paper presents a comprehensive survey of the decomposition-based MOEAs proposed in the last decade.

Proximal Point Algorithms for Multi-criteria Optimization with the Difference of Convex Objective Functions

This paper focuses on solving a class of multi-criteria optimization with the difference of convex objective functions. Proximal point algorithms, extensively studied for scalar optimization, are extended to our setting. We show that the proposed algorithms are well posed and globally convergent to a critical point. For an application, the new methods are used to a multi-criteria model arising in portfolio optimization. The numerical results show the efficiency of our methods.

Efficiency for a generalized form of vector optimization problems in complex space

A generalized form of vector optimization problems in complex space is considered, where both the real and the imaginary parts of the objective functions are taken into account. The efficient solutions are defined and characterized in terms of optimal solutions of related appropriate scalar optimization problems. These scalar problems are formulated by means of vectors in the dual of the domination cone. Under analyticity hypotheses about the functions, complex extensions to necessary and sufficient conditions for efficiency of Kuhn–Tucker type are established. Most of the corresponding results of previous studies (in both finite-dimensional complex and real spaces) can be recovered as particular cases.

Robustness Testing of Model Based Multiple Criteria Decisions: Fundamentals and Applications

Robustness or insensitivity is a desirable property of decisions; however, most texts on robustness and/or sensitivity analysis do not define it precisely. A broad literature in this field concentrates on robust design of decisions (including robust optimization). This paper focuses on robustness testing, that is, checking whether a design has actually resulted in robust properties of the system if some of basic assumptions are changed. We propose a general framework of such robustness testing and show that robustness is a property of the relation between three (classes of) models: a basic model of the decision situation, a second model of possible perturbations of the first model, and a model of implementation of the decision, optionally taking into account some measurements of the impact of perturbations. Typical approaches to robustness or sensitivity analysis assume tacitly that the first two models can be combined and analyze parameters deviations in one combined model. However, the role of the first two models can be asymmetric if some optimization of the decision is performed on the first model. We extend this framework, intended originally for single criteria (scalar) optimization to multiple criteria (vector) optimization. The proposed approach is illustrated by diverse examples.

Semivectorial Bilevel Optimization on Riemannian Manifolds

In this paper, we deal with the semivectorial bilevel problem in the Riemannian setting. The upper level is a scalar optimization problem to be solved by the leader, and the lower level is a multiobjective optimization problem to be solved by several followers acting in a cooperative way inside the greatest coalition and choosing among Pareto solutions with respect to a given ordering cone. For the so-called optimistic problem, when the followers choice among their best responses is the most favorable for the leader, we give optimality conditions. Also for the so-called pessimistic problem, when there is no cooperation between the leader and the followers, and the followers choice may be the worst for the leader, we present an existence result.

Pointwise well-posedness and scalarization in set optimization

In this paper, some notions of pointwise well-posedness for set optimization problems are introduced. Some relationships among these notions are established. Using a new nonlinear scalarization function, pointwise well-posed set optimization problems are characterized by means of a family of Tykhonov well-posed scalar optimization problems. Also, three classes of well-posed set optimization problems are identified.

Decomposing the user-preference in multiobjective optimization

Preference information (such as the reference point) of the decision maker (DM) is often used in multiobjective optimization; however, the location of the specified reference point has a detrimental effect on the performance of multiobjective evolutionary algorithms (MOEAs). Inspired by multiobjective evolutionary algorithm-based decomposition (MOEA/D), this paper proposes an MOEA to decompose the preference information of the reference point specified by the DM into a number of scalar optimization subproblems and deals with them simultaneously (called MOEA/D-PRE). This paper presents an approach of iterative weight to map the desired region of the DM, which makes the algorithm easily obtain the desired region. Experimental results have demonstrated that the proposed algorithm outperforms two popular preference-based approaches, g-dominance and r-dominance, on continuous multiobjective optimization problems (MOPs), especially on many-objective optimization problems. Moreover, this study develops distinct models to satisfy different needs of the DM, thus providing a new way to deal with preference-based multiobjective optimization. Additionally, in terms of the shortcoming of MOEA/D-PRE, an improved MOEA/D-PRE that dynamically adjusts the size of the preferred region is proposed and has better performance on some problems.

A SUBGRADIENT METHODS FOR NON-SMOOTH VECTOR OPTIMIZATION PROBLEMS

Vector optimization problems are a significant extension of scalar optimization and have wide range of application in various fields of economics, decision theory, game theory, information theory and optimal control theory. In this paper, unlike general subgradient methods, bundle methods and gradient sampling methods used to solve nonsmooth vector optimization problems which include scalarization approach, a subgradient method without usual scalariza- tion approached is proposed for minimizing a non-differentiable convex func- tion which works directly with vector-valued function. A general sub-gradient method for non-smooth convex optimization that includes regularization and interior point variants of Newton's Method are proposed. This algorithm builds a sequence of efficient points at the interior of epigraph of objective function which satisfies KKT conditions. In this paper, under the suitable conditions it is proved that the sequence generated by algorithm converges to ǫ-efficient point.

A new algorithm for linearly constrained c-convex vector optimization with a supply chain network risk application

We study a class of vector optimization problems with a C-convex objective function under linear constraints. We extend the proximal point algorithm used in scalar optimization to vector optimization. We analyze both the global and local convergence results for the new algorithm. We then apply the proximal point algorithm to a supply chain network risk management problem under bi-criteria considerations.