Nonconvex Feasible Region Research Articles

Convergence error analysis of reflected gradient Langevin dynamics for non-convex constrained optimization

AbstractGradient Langevin dynamics and a variety of its variants have attracted increasing attention owing to their convergence towards the global optimal solution, initially in the unconstrained convex framework while recently even in convex constrained non-convex problems. In the present work, we extend those frameworks to non-convex problems on a non-convex feasible region with a global optimization algorithm built upon reflected gradient Langevin dynamics and derive its convergence rates. By effectively making use of its reflection at the boundary in combination with the probabilistic representation for the Poisson equation with the Neumann boundary condition, we present promising convergence rates, particularly faster than the existing one for convex constrained non-convex problems.

A Novel Model-Free Deep Reinforcement Learning Framework for Energy Management of a PV Integrated Energy Hub

This paper utilizes a fully model-free and data-driven deep reinforcement learning (DRL) framework to develop an intelligent controller that can exploit information to optimally schedule the energy hub with the aim of minimizing energy costs and emissions. By posing the energy hub scheduling problem as a multi-dimensional continuous state and action space, the proposed deep deterministic policy gradient (DDPG) method enables more cost-effective control strategies. The method can lead to a more efficient operation by considering nonlinear physical characteristics of the energy hub components like nonconvex feasible operating regions of combined heat and power (CHP) units, valve-point effects of power-only units, and fuel cell dynamic efficiency. Moreover, to provide great potential for the DDPG agent to learn an optimal policy in an efficient way, a hybrid forecasting model based on convolutional neural networks (CNNs) and bidirectional long short-term memories (BLSTMs) is developed to overcome the risk associated with PV power generation that can be highly intermittent, particularly on cloudy days. The effectiveness and applicability of the proposed scheduling framework in reducing energy costs and emissions while coping with uncertainties are demonstrated by comparing it against conventional robust optimization and stochastic programming approaches as well as state-of-the-art DRL methods in different case studies.

Multi-objective optimization of a lithium cell design operating in a cyclic process

Multi-objective optimization frameworks of lithium-ion whole-cell design operating in a cyclic process are presented in this paper. The objective functions involved are maximization of capacity, maximization of specific energy in discharge and maximization of cycle energy efficiency. The considered design variables are electrode thicknesses, solid volume fraction and porosity of electrode compartments, and cell mass per unit cross-sectional area. In addition, the constant current value of charge and discharge is optimized. A phenomenological mathematical model developed in GAMS (General Algebraic Modeling System) environment is applied to achieve the stated goals and the multi-objective problems are solved using an epsilon constraint method-based, which is effective for finding solutions in non-convex feasible regions. Solutions for two different cell chemistries are analyzed: LMO and LCO. Pareto curves and surfaces are obtained and sets of solutions with the highest contradictions and with the most balanced agreement between the objectives involved are identified. When comparing optimal LMO designs with optimal LCO designs, the former result on average 48 % less capacity, 19 % less specific energy and 31 % more energy efficiency. Results show that material type-dependent variables such as active solids have a strong influence on the specific energy and cycle energy efficiency. While the electrode thicknesses show a general trend regardless of the type of chemistry, presenting a strong influence on cell capacity values.

Unified Description and Optimization Method of Dual Active Bridge DC–DC Converters

The optimization of dual active bridge converters (DAB) is widely studied and numerous modulation schemes are proposed to enhance the performance of a DAB converter. However, a unified description method both suitable for time and frequency domain analysis still lacks. Besides, the optimization process encounters a difficulty of the potential nonconvex feasible region restricted by the power transmission equality, making the regular convex optimization methods no longer applicable. To solve these problems, this article gave a unified description method both suitable for time and frequency domain analysis based on the standard decomposition of instantaneous electric quantities. Besides, a unified optimization method is proposed based on tangent point searching for the feasible region and optimization objective surfaces. With the proposed unified optimization method, the optimization objective can be designed arbitrarily as long as it is piecewise continuous, and smooth. An efficiency-oriented optimization objective design method is introduced considering all electric quantities as well as the soft-switching conditions. Finally, the experimental results confirm the unified optimization method and the efficiency-oriented optimization objective design method.

Adaptive Hammerstein Filtering via Recursive Non-Convex Projection

Hammerstein model with a static nonlinearity followed by a linear filter is commonly used in numerous applications. This paper focuses on adaptive filtering techniques for parameter identification of Hammerstein systems and output prediction of nonlinear systems. By formulating the underlying filtering problem as a recursive bilinear least-squares optimization with the non-convex feasible region constraint, we develop a recursive non-convex projected least-squares (RncPLS) algorithm based on alternating direction method of multipliers (ADMM). The RncPLS algorithm alternates between implementing ridge regression and projecting on the non-convex feasible set, which successively refines the system parameters. The convergence and accuracy properties of the proposed RncPLS algorithm are theoretically investigated. Moreover, extensive simulation results in the context of system identification, nonlinear predication, and acoustic echo cancellation, are also included to demonstrate the performance characteristics of the proposed algorithm.

Robust 3D-Trajectory and Time Switching Optimization for Dual-UAV-Enabled Secure Communications

This paper investigates a dual-unmanned aerial vehicle (UAV)-enabled secure communication system, in which, a UAV moves around to send confidential messages to a mobile user while another cooperative UAV transmits artificial noise signals to confuse malicious eavesdroppers. Both UAVs have energy constraints and the location information of eavesdroppers is imperfect. We consider a worst-case secrecy rate maximization problem of the mobile user over all time slots. This optimization problem is solved by jointly designing the three-dimensional (3D) trajectory of UAVs and the time allocation (recharging and service or jamming) under practical constraints including maximum UAV speed, UAV collision avoidance, UAV positioning error, and UAV energy harvesting. Specifically, we adopt a more practical UAV-ground channel model with both large-scale and small-scale fading components. Due to the non-convex feasible region constructed by the complicated constraints, directly finding the optimal solution of the original problem is intractable. To address this issue, we decouple the original optimization problem into three subproblems and develop an iterative algorithm to find its suboptimal solution by using the block coordinate descent technique. To solve each subproblem, certain advanced optimization tools, such as integer relaxation, S-procedure, and successive convex approximation techniques, are utilized. Numerical simulation results are provided to corroborate the theoretical derivations and to evaluate the performance of the proposed algorithm. Additionally, the numerical results assist to draw new insights on the 3D UAV trajectory by comparing the performance with conventional two-dimensional (2D) schemes.

An Innovative Hybrid Heap-Based and Jellyfish Search Algorithm for Combined Heat and Power Economic Dispatch in Electrical Grids

This paper proposes a hybrid algorithm that combines two prominent nature-inspired meta-heuristic strategies to solve the combined heat and power (CHP) economic dispatch. In this line, an innovative hybrid heap-based and jellyfish search algorithm (HBJSA) is developed to enhance the performance of two recent algorithms: heap-based algorithm (HBA) and jellyfish search algorithm (JSA). The proposed hybrid HBJSA seeks to make use of the explorative features of HBA and the exploitative features of the JSA to overcome some of the problems found in their standard forms. The proposed hybrid HBJSA, HBA, and JSA are validated and statistically compared by attempting to solve a real-world optimization issue of the CHP economic dispatch. It aims to satisfy the power and heat demands and minimize the whole fuel cost (WFC) of the power and heat generation units. Additionally, a series of operational and electrical constraints such as non-convex feasible operating regions of CHP and valve-point effects of power-only plants, respectively, are considered in solving such a problem. The proposed hybrid HBJSA, HBA, and JSA are employed on two medium systems, which are 24-unit and 48-unit systems, and two large systems, which are 84- and 96-unit systems. The experimental results demonstrate that the proposed hybrid HBJSA outperforms the standard HBA and JSA and other reported techniques when handling the CHP economic dispatch. Otherwise, comparative analyses are carried out to demonstrate the suggested HBJSA’s strong stability and robustness in determining the lowest minimum, average, and maximum WFC values compared to the HBA and JSA.

Scheduling heat and power microgrids with storage systems, photovoltaic, wind, geothermal power units and solar heaters

Combined heat and power (CHP) microgrids (MGs) are a set of CHP units, boilers, power-only distributed generation (DG) units and storage systems that simultaneously supply heat and power demand. In this research, the objective is to provide a comprehensive mixed integer linear programming (MILP) model for unit commitment (UC) in CHP MGs including fossil-fueled power-only DGs, boilers, CHP units, photovoltaic, wind and geothermal power units, solar heater, battery charging station (BCS), adjustable thermal loads, battery energy storage (BES) and thermal energy storage (TES) units. In the proposed model, all operational constraints of CHP units, boilers and power-only units such as minimum up/down time and start-up/ shut-down ramp rate limits are considered and the commitment history of generation units prior to operation horizon is considered. In the proposed model, type II CHPs with non-convex feasible operating region (FOR) are used and their non-convex FOR is transformed into two convex FORs. The proposed model is solved with CPLEX solver in general algebraic modeling system (GAMS) which guarantees the achievement of the global optimum. The model is solved for different scenarios wherein the effect of BCS, effect of storage systems as well as the effect of adjustability of loads on MG operation is investigated. According to the results, BES and TES respectively decrease MG operation cost by 7.4% and 1.82%.and BCS adds 20.19% to the MG operation cost. The results also show that with replacement of adjustable thermal loads with evenly distributed loads, MG operation cost increases by 0.6%.

A robust algorithm for global optimization problems

In this paper, a global optimization algorithm namely Kerk and Rohanin’s Trusted Region is used to find the global minimizers by employing an interval technique; with it, the algorithm can find the region where a minimizer is located and will not get trapped in a local one. It is able to find the convex part within the non-convex feasible region. This algorithm has descent property and global convergence. The numerical results have shown the algorithm has an outstanding capability in locating global minimizers.

Bounds on mean absolute deviation portfolios under interval-valued expected future asset returns

This work concerns a suitable range of optimal portfolio compositions as well as their optimal returns in the mean absolute deviation portfolio selection model when a threshold return expected from the investment is given. Disagreement in measurements of expected future asset returns is resolved by the alternative representation of their intervals. The nonlinear behavior of the resultant parametric model with its nonconvex feasible region is discussed in detail. Constraints used in achieving exact bounds can be relaxed for faster computation and more accurate results, leading to relaxed bounds which can be improved by basis stability. This is particularly useful for a securities company to create more appealing advertisement on its financial products and also for an investor to screen out unfavorable assets from a variety of instruments to spend less time on and reduce expenses incurred in the process of fundamental and technical analysis. The relaxed bounds are compared with the bounds obtained by bilevel and nonlinear approaches.

Optimal design of electrical machines assisted by hybrid surrogate model based algorithm

In this paper, for design of large-scale electromagnetic problems, a novel robust global optimization algorithm based on surrogate models is presented. The proposed algorithm can automatically select a proper meta-model technique among multiple alternatives. In this paper, three representative meta-modeling techniques including ordinary Kriging, universal Kriging, and response surface method with multi-quadratic radial basis functions are applied. In each optimization iteration, the above three models are used for parallel calculation. The proposed hybrid surrogate model optimization algorithm synthesizes advantages of these different meta-models. Without verification of a specific meta-model, a suitable one for the engineering problem to be analyzed is automatically selected. Therefore, the proposed algorithm intends to make a better trade-off between numerical efficiency and searching accuracy for solving engineering problems, which are characterized by stronger non-linearity, higher complexity, non-convex feasible region, and expensive performance analysis.

Determining the Optimal Operating Point of CHP Units with Nonconvex Characteristics in the Context of Combined Heat and Power Scheduling Problem

ABSTRACTDetermining the optimal operating point of generating units in the optimal combined heat and power scheduling is an economic optimization problem aimed at minimizing the total operating costs. The generation cost would nonlinearly increase with the increase in the power generation of fossil fuel thermal units. On the other hand, in combined heat and power (CHP) units, the amounts of power and heat generation are highly mutually dependent. Therefore, the determination of the optimal operating points of such units would be much more difficult compared to electric power or heat-only generating units. Obviously, the aim beyond implementing the day-ahead generation scheduling (GS) problem is to determine the optimal operating points of generating units. In this regard, the power outputs of the units are specified such that the total operating costs of the system are minimized taking into consideration the technical and economic constraints. This paper solves the GS problem in the presence of CHP units with nonconvex feasible operating regions (FORs). This paper investigates the problem of optimal operating point determination of CHP units along with electrical and heat-only units.

A Branch–Bound Cut Technique for Non-linear Fractional Multi-objective Optimization Problems

This article establishes a branch–bound technique to solve nonlinear convex–convex fractional multi-objective optimization problem in the non-convex feasible region. As far as the authors are concerned, this kind of problem is not solved by any other author in the literature. By transformation, multi-objective non-linear fractional problem is transformed into a multi-objective non-linear optimization problem. After giving preferences of weight to each objective, the original NLFMOOP is transformed into a nonlinear single-objective programming problem. Lagrange’s theorem of weak duality is used to find lower and upper bound for single objective nonlinear optimization problems in the feasible region. Some theoretical results for solving the multi-objective non-linear fractional problem have also been established. For showing the application of the proposed method, it has been applied to two numerical problems.

A Two-Stage Mathematical Programming Approach for the Solution of Combined Heat and Power Economic Dispatch

The optimal generation scheduling of combined heat and power units aims to minimize total cost of generation units satisfying power and heat demands.The combined heat and power economic dispatch problem should be solved considering a series of electrical and operational constraints, which will be more complicated considering valve-point effects of power-only units and nonconvex feasible operating regions of cogeneration units. In such a case, the problem has nonconvex and nondifferential characteristic. In this article, a two-stage model is proposed to handle the nonconvexity and nondifferentiability of valve-point effects in cost functions of power-only units. The proposed model obtains a convex feasible operating region in the first stage using a linear approximation model. In addition, an equivalent formulation is proposed in the second stage to handle the nondifferentiable term of valve-point effects using the obtained convex feasible operating regions. The proposed model is implemented on two case studies to evaluate performance of the proposed model.

From Obstacle-Based Space Partitioning to Corridors and Path Planning. A Convex Lifting Approach

This letter proposes a novel methodology for path generation in known and congested multi-obstacle environments. Our aim is to solve an open problem in navigation within such environments: the feasible space partitioning in accordance with the distribution of obstacles. It is shown that such a partitioning is a key concept toward the generation of a corridor in cluttered environments. Once a corridor between an initial and a final position is generated, the selection of a path is considerably simplified in comparison with the methods which explore the original non-convex feasible regions of the environment. The core of the methodology presented here is the construction of a convex lifting which boils down to a convex optimization. This letter covers both the mathematical foundations and the computational details of the implementation and aims to illustrate the concepts with geometrical examples.

Generating $\alpha $-dense curves in non-convex sets to solve a class of non-smooth constrained global optimization

This paper deals with the dimensionality reduction approach to study multi-dimensional constrained global optimization problems where the objective function is non-differentiable over a general compact set $D$ of $\mathbb{R}^{n}$ and H\{o}lderian. The fundamental principle is to provide explicitly a parametric representation $x_{i}=\ell _{i}(t),1\leq i\leq n$ of $\alpha $-dense curve $\ell_{\alpha }$ in the compact $D$, for $t$ in an interval $\mathbb{I}$ of $\mathbb{R}$, which allows to convert the initial problem to a one dimensional H\{o}lder unconstrained one. Thus, we can solve the problem by using an efficient algorithm available in the case of functions depending on a single variable. A relation between the parameter $\alpha $ of the curve $\ell _{\alpha }$ and the accuracy of attaining the optimal solution is given. Some concrete $\alpha $ dense curves in a non-convex feasible region $D$ are constructed. The numerical results show that the proposed approach is efficient.

Computing a Pessimistic Stackelberg Equilibrium with Multiple Followers: The Mixed-Pure Case

The search problem of computing a Stackelberg (or leader-follower)equilibrium (also referred to as an optimal strategy to commit to) has been widely investigated in the scientific literature in, almost exclusively, the single-follower setting. Although the optimistic and pessimistic versions of the problem, i.e., those where the single follower breaks any ties among multiple equilibria either in favour or against the leader, are solved with different methodologies, both cases allow for efficient, polynomial-time algorithms based on linear programming. The situation is different with multiple followers, where results are only sporadic and depend strictly on the nature of the followers’ game. In this paper, we investigate the setting of a normal-form game with a single leader and multiple followers who, after observing the leader’s commitment, play a Nash equilibrium. When both leader and followers are allowed to play mixed strategies, the corresponding search problem, both in the optimistic and pessimistic versions, is known to be inapproximable in polynomial time to within any multiplicative polynomial factor unless textsf {P}=textsf {NP}. Exact algorithms are known only for the optimistic case. We focus on the case where the followers play pure strategies—a restriction that applies to a number of real-world scenarios and which, in principle, makes the problem easier—under the assumption of pessimism (the optimistic version of the problem can be straightforwardly solved in polynomial time). After casting this search problem (with followers playing pure strategies) as a pessimistic bilevel programming problem, we show that, with two followers, the problem is NP-hard and, with three or more followers, it cannot be approximated in polynomial time to within any multiplicative factor which is polynomial in the size of the normal-form game, nor, assuming utilities in [0, 1], to within any constant additive loss stricly smaller than 1 unless textsf {P}=textsf {NP}. This shows that, differently from what happens in the optimistic version, hardness and inapproximability in the pessimistic problem are not due to the adoption of mixed strategies. We then show that the problem admits, in the general case, a supremum but not a maximum, and we propose a single-level mathematical programming reformulation which asks for the maximization of a nonconcave quadratic function over an unbounded nonconvex feasible region defined by linear and quadratic constraints. Since, due to admitting a supremum but not a maximum, only a restricted version of this formulation can be solved to optimality with state-of-the-art methods, we propose an exact ad hoc algorithm (which we also embed within a branch-and-bound scheme) capable of computing the supremum of the problem and, for cases where there is no leader’s strategy where such value is attained, also an alpha -approximate strategy where alpha > 0 is an arbitrary additive loss (at most as large as the supremum). We conclude the paper by evaluating the scalability of our algorithms via computational experiments on a well-established testbed of game instances.

Coordinated Operation of Electricity and Natural Gas Systems: A Convex Relaxation Approach

The variability in the generation dispatch of the natural gas generation units will lead to fluctuation in natural gas demand profile that could further jeopardize the security of the natural gas network. The coordinated operation of electricity and natural gas infrastructure systems would help to improve the security and reliability measures in both infrastructure systems and mitigate the risk of demand curtailment. The electricity and natural gas network operation problems are non-convex mixed-integer nonlinear programming problems that are hard to solve in polynomial time. The non-convex feasible regions are formed by the Weymouth constraint and the introduced binary commitment decision variables in the natural gas and electricity network operation problems, respectively. This paper utilized a sparse semidefinite programming (SDP) relaxation to procure the optimal solution for the coordinated operation of electricity and natural gas networks. The presented algorithm leverages the sparseness of the natural gas network to construct several small matrices of lifting variables that are used to form a tight and traceable SDP relaxation. A set of valid constraints that tighten the relaxation ensures the exactness of the solution procured from the relaxed problem. The effectiveness of the presented approach is shown in case studies.

Linearization of Euclidean norm dependent inequalities applied to multibeam satellites design

Euclidean norm computations over continuous variables appear naturally in the constraints or in the objective of many problems in the optimization literature, possibly defining non-convex feasible regions or cost functions. When some other variables have discrete domains, it positions the problem in the challenging Mixed Integer Nonlinear Programming (MINLP) class. For any MINLP where the nonlinearity is only present in the form of inequality constraints involving the Euclidean norm, we propose in this article an efficient methodology for linearizing the optimization problem at the cost of entirely controllable approximations even for non convex constraints. They make it possible to rely fully on Mixed Integer Linear Programming and all its strengths. We first empirically compare this linearization approach with a previously proposed linearization approach of the literature on the continuous k-center problem. This methodology is then successfully applied to a critical problem in the telecommunication satellite industry: the optimization of the beam layouts in multibeam satellite systems. We provide a proof of the NP-hardness of this very problem along with experiments on a realistic reference scenario.

Mixed-integer nonlinear programming framework for combined heat and power units with nonconvex feasible operating region: Feasibility, optimality, and flexibility evaluation

This paper presents an optimization framework to determine the optimal operating points of combined heat and power (CHP) units with nonlinear, nonconvex feasible operating region (FOR). The mentioned problem is the economic dispatch (ED) of heat-only units, thermal units, and CHP units. Also, the electric units are of thermal technology while their valve-point impact is taken into consideration. Also, the heat power curve of heat-only units is nonlinear. It is noted that the FOR of CHP units has been defined both as convex and nonconvex regions. For those with nonconvex characteristic, a method is proposed to convert it into a convex characteristic. Accordingly, a separate binary variable must be defined to determine the optimal operating point in each convex region. Thus, the presented problem in this paper is of mixed-integer nonlinear programming (MINLP) type modeled in General Algebraic Modeling System (GAMS) software. In this respect, different case studies have been presented to assess the optimality, feasibility, and the flexibility of the model. It is noteworthy that the valve-point effect of thermal units and different FORs of CHP units as well as the electrical power losses have been considered.