Random Optimization Problems Research Articles

Multi-objective topology optimization method for multi-axis random vibration based on hybrid cellular automata

Considering the lack of effective solutions for multi-axis random vibration topology optimization problems, and recognizing that multi-axis random vibration excitation is the most common loading conditions experienced by structures during their operational life, this paper proposes a multi-objective topology optimization method for multi-axis random vibration. By combining Hybrid Cellular Automata (HCA) and Analytic Hierarchy Process (AHP), this method aims to enhance the vibration resistance of structures and extend their fatigue life during the topology optimization phase. To validate the effectiveness and engineering practicality of this method, two engineering cases were designed: a cantilever beam case and an automotive steering knuckle case. Multi-objective topology optimization were conducted with objectives including mass, stiffness, and vibration intensity metric (root-mean-square stress). The two cases demonstrate that the control group's topology model exhibits weaker resistance to random vibrations and shorter fatigue life. In contrast, the validation group's topology model introduces an additional load path near the excitation point, effectively dissipating the impact of vibration excitation. This enhances the structure's vibration resistance and significantly extends its fatigue life.

Notes on random optimal control equilibrium problem via stochastic inverse variational inequalities

The main objective of the paper is to analyze how policymakers influence the random oligopolistic market equilibrium problem. To this purpose, random optimal control equilibrium conditions are introduced. Since the random optimal regulatory tax is characterized by a stochastic inverse variational inequality, existence and well-posedness results on such an inequality are proved. At last a numerical example is discussed.

The existence of random solutions to random optimization problems

Abstract In this paper, random optimization problems are investigated. Some sufficient conditions ensuring the existence of random solutions to random optimization problems are proposed.

Hardness of Random Optimization Problems for Boolean Circuits, Low-Degree Polynomials, and Langevin Dynamics

Hardness of Random Optimization Problems for Boolean Circuits, Low-Degree Polynomials, and Langevin Dynamics

Closest-vector problem and the zero-temperature p-spin landscape for lossy compression.

We consider a high-dimensional random constrained optimization problem in which a set of binary variables is subjected to a linear system of equations. The cost function is a simple linear cost, measuring the Hamming distance with respect to a reference configuration. Despite its apparent simplicity, this problem exhibits a rich phenomenology. We show that different situations arise depending on the random ensemble of linear systems. When each variable is involved in at most two linear constraints, we show that the problem can be partially solved analytically, in particular we show that upon convergence, the zero-temperature limit of the cavity equationsreturns the optimal solution. We then study the geometrical properties of more general random ensembles. In particular we observe a range in the density of constraints at which the system enters a glassy phase where the cost function has many minima. Interestingly, the algorithmic performances are only sensitive to another phase transition affecting the structure of configurations allowed by the linear constraints. We also extend our results to variables belonging to GF(q), the Galois field of order q. We show that increasing the value of q allows to achieve a better optimum, which is confirmed by the replica-symmetric cavity method predictions.

High-dimensional optimization under nonconvex excluded volume constraints.

We consider high-dimensional random optimization problems where the dynamical variables are subjected to nonconvex excluded volume constraints. We focus on the case in which the cost function is a simple quadratic cost and the excluded volume constraints are modeled by a perceptron constraint satisfaction problem. We show that depending on the density of constraints, one can have different situations. If the number of constraints is small, one typically has a phase where the ground state of the cost function is unique and sits on the boundary of the island of configurations allowed by the constraints. In this case, there is a hypostatic number of marginally satisfied constraints. If the number of constraints is increased one enters a glassy phase where the cost function has many local minima sitting again on the boundary of the regions of allowed configurations. At the phase transition point, the total number of marginally satisfied constraints becomes equal to the number of degrees of freedom in the problem and therefore we say that these minima are isostatic. We conjecture that by increasing further the constraints the system stays isostatic up to the point where the volume of available phase space shrinks to zero. We derive our results using the replica method and we also analyze a dynamical algorithm, the Karush-Kuhn-Tucker algorithm, through dynamical mean-field theory and we show how to recover the results of the replica approach in the replica symmetric phase.

Optimal control for multistage uncertain random dynamic systems with multiple time delays

A complex system exhibiting uncertainty as well as randomness is an uncertain random dynamic system. A novel idea of an uncertain random matrix is introduced as an interaction between uncertainty and randomness. Thereby a time-delayed uncertain random dynamic system is formulated. And then the corresponding time-delayed optimal control problem is first presented in this paper. To tackle a problem like that, a method is proposed to translate it into an equivalent optimization problem without time delay. The relevant recursion equations are developed with the dynamic programming technique to solve the converted optimization problem. Solving the related recursion equations yields the answer to such an optimization problem. The presented approach offers a unified framework for tackling the discrete-time uncertain random optimal control problem involving time delays. In this framework, three algorithms are successfully devised in doing with time-delayed optimal control problems for dynamic systems regarding linear, quadratic, and cubic controls. Meanwhile, the optimal solutions for numerical examples are found by the raised algorithms. Finally, a problem of optimal control for a two-spool turbofan engine is investigated as an application of our results.

Central limit theorems for combinatorial optimization problems on sparse Erdős–Rényi graphs

For random combinatorial optimization problems, there has been much progress in establishing laws of large numbers and computing limiting constants for the optimal values of various problems. However, there has not been as much success in proving central limit theorems. This paper introduces a method for establishing central limit theorems in the sparse graph setting. It works for problems that display a key property which has been variously called “endogeny,” “long-range independence” and “replica symmetry” in the literature. Examples of such problems are maximum weight matching, λ-diluted minimum matching, and optimal edge cover.

Uncertain random mean–variance–skewness models for the portfolio optimization problem

In the face of complex financial phenomena, describing the high uncertainties in financial markets remains a challenging issue in modelling and decision making. In this paper, chance theory is used to analyse the hybrid uncertainty that combines random returns and uncertain returns. We regard the total return as an uncertain random variable and study the uncertain random portfolio optimization problem. We first define the skewness of an uncertain random variable and derive some important properties and explicit expressions with deterministic distributions. These theoretical results help transform the model into a deterministic form. Then, uncertain random mean–variance–skewness portfolio optimization models and their corresponding equivalents are established to meet the diverse needs of investors. Finally, we provide two numerical experiments to illustrate the applicability of the proposed model, one of which demonstrates that the model can be applied in the area of energy finance. It is shown that when uncertain asset returns are asymmetric, this model can be used to make investment decisions for investors. We also find that a higher return is accompanied by higher risk and higher skewness.

Integrated energy system planning research based on big data load prediction method

The planning of integrated energy system is a very complex multi-target, multi-constraint, nonlinear, random uncertainty mixed integrated combination optimization problem, its planning and design process should not only consider the interdependence between the system capacity, energy conversion, energy storage, energy use and other links, but also consider the interaction and integration of cold, hot, electricity and other multi-energy flows, which is essentially a non-deterministic polynomial difficult problem. China’s energy continues to develop rapidly, all kinds of sensors and intelligent equipment data is increasing, the data obtained in the equipment and all kinds of sensors collected energy load prediction related factors such as temperature, weather, wind speed and other data volume increased dramatically, the data dimension is also increasing, the scale of data has also increased from GB to TB or even higher, based on the traditional prediction methods and intelligent prediction methods, has been far below the load forecast desired to achieve accuracy and speed requirements, Therefore, the use of big data technology to predict energy demand is an important future direction.

Integrated energy system planning study based on load prediction

The planning of integrated energy system is a very complex multi-objective, multi-constraint, nonlinear, random uncertain hybrid combination optimization problem, its planning and design process should consider not only the system capacity, energy exchange, energy storage, energy and other links between the interdependence, but also the interaction and mixing of cold, hot, electricity and other multi-energy flow, which is essentially a non-deterministic polynomial problem. Based on load prediction technology, combined with scene generation, multi-interconnected energy system modeling and other technologies, around the integrated energy system planning and design, consider the comprehensive evaluation of the whole life cycle, an optimal configuration of the integrated energy system is formed.

Persistence-Speed Coupling Enhances the Search Efficiency of Migrating Immune Cells.

Migration of immune cells within the human body allows them to fulfill their main function of detecting pathogens. We present experimental evidence showing the optimality of the search strategy of these cells, which is of crucial importance to achieve an efficient immune response. We find that the speed and directional persistence of migrating dendritic cells in our invitro experiments are highly correlated, which enables them to reduce their search time. We introduce theoretically a new class of random search optimization problems by minimizing the mean first-passage time (MFPT) with respect to the strength of the coupling between influential parameters. We derive an analytical expression for the MFPT in a confined geometry and verify that the correlated motion enhances the search efficiency if the mean persistence length is sufficiently shorter than the confinement size. Our correlated search optimization approach provides an efficient searching recipe and predictive power in a broad range of correlated stochastic processes.

Optimal stopping for measure-valued piecewise deterministic Markov processes

AbstractThis paper investigates the random horizon optimal stopping problem for measure-valued piecewise deterministic Markov processes (PDMPs). This is motivated by population dynamics applications, when one wants to monitor some characteristics of the individuals in a small population. The population and its individual characteristics can be represented by a point measure. We first define a PDMP on a space of locally finite measures. Then we define a sequence of random horizon optimal stopping problems for such processes. We prove that the value function of the problems can be obtained by iterating some dynamic programming operator. Finally we prove via a simple counter-example that controlling the whole population is not equivalent to controlling a random lineage.

Random Caching Optimization in Large-Scale Cache-Enabled Internet of Things Networks

The introduction of cache can reduce unnecessary traffic load and improve latency in the wireless access networks, especially for wireless video broadcasting. But how cache impacts video broadcasting with scalable video coding (SVC) is still an open problem. In this paper, we analyze the optimization of random caching in a large-scale cache-enabled Internet of Things networks. Specifically, we propose to optimize the random caching strategy that aims to maximize the successful transmission probability (STP) of the video contents at edge base stations (BSs). To this end, by using the stochastic geometry theory, we derive analytical expression of STP by considering SVC to satisfy different levels of quality of service requirements. We develop a gradient-based iterative algorithm to search the local optimal solution for the general random caching strategy optimization problem. The asymptotical optimal caching strategy is obtained with a lower complexity. The closed-form STP is also obtained in high signal-to-noise ratio and particular cache size. Based on the closed-form STP expressions, the random caching strategy, i.e., caching probability of different video contents, is further optimized to enhance STP performance. Compare to different reference schemes, the proposed caching strategy improves the STP up to 18% and 21.6% with the low density of BSs and the high density of BSs, respectively.

On sparsity of the solution to a random quadratic optimization problem

The standard quadratic optimization problem (StQP), i.e. the problem of minimizing a quadratic form $$\mathbf{x}^TQ\mathbf{x}$$ on the standard simplex $$\{\mathbf{x}\ge \mathbf{0}: \mathbf{x}^T\mathbf{e}=1\}$$ , is studied. The StQP arises in numerous applications, and it is known to be NP-hard. Chen, Peng and Zhang showed that almost certainly the StQP with a large random matrix $$Q=Q^T$$ , $$Q_{i,j},\, (i\le j)$$ being independent and identically concave-distributed, attains its minimum at a point $$\mathbf{x}$$ with support size $$|\{j: x_j>0\}|$$ bounded in probability. Later Chen and Peng proved that for $$Q=(M+M^T)/2$$ , with $$M_{i,j}$$ i.i.d. normal, the likely support size is at most 2. In this paper we show that the likely support size is poly-logarithmic in n, the problem size, for a considerably broader class of the distributions. Unlike the cited papers, the mild constraints are put on the asymptotic behavior of the distribution at a single left endpoint of its support, rather than on the distribution’s shape elsewhere. It also covers the distributions with the left endpoint $$-\infty $$ , provided that the distribution of $$Q_{i,j},\, (i\le j)$$ (of $$M_{i,j}$$ , if $$Q=(M+M^T)/2$$ resp.) has a (super/sub) exponentially narrow left tail.

RANDOM COMBINATORIAL OPTIMIZATION PROBLEMS: MEAN FIELD AND FINITE-DIMENSIONAL RESULTS

This PhD thesis is organized as follows. In the first two chapters I will review some basic notions of statistical physics of disordered systems, such as random graph theory, the mean-field approximation, spin glasses and combinatorial optimization. The replica method will also be introduced and applied to the Sherrington-Kirkpatrick model, one of the simplest mean-field models of spin-glasses. The second part of the thesis deals with mean-field combinatorial optimization problems. The attention will be focused on the study of finite-size corrections of random integer matching problems (chapter 3) and fractional ones (chapter 4). In chapter 5 I will discuss a very general relation connecting multi-overlaps and the moments of the cavity magnetization distribution. In the third part we consider random Euclidean optimization problems. I will start solving the traveling-salesman-problem (TSP) in one dimension both in its bipartite and monopartite version (chapter 6). In chapter 7 I will discuss the possible optimal solutions of the 2-factor problem. In chapter 8 I will solve the bipartite TSP in two dimensions, in the limit of large number of points. Chapter 9 contains some conclusions.

Uncertain random quadratic bottleneck assignment problem

Uncertain random expected value simulation is one of the most important techniques to solve uncertain random optimization problems where random variables coexist with uncertain variables. A general simulation algorithm is designed to estimate the uncertain random expected value. Furthermore, an uncertain random quadratic bottleneck assignment problem is proposed and an uncertain random expected value model is presented. An algorithm is designed to find a lower bound of the uncertain random quadratic bottleneck assignment problem.

Uncertain random simulation algorithm with application to bottleneck assignment problem

Uncertain random simulation plays an important role in solving uncertain random optimization problems that include random variables and uncertain variables. In this paper, an uncertain random simulation is proposed and developed to obtain the chance distribution, $$\alpha $$ -pessimistic value and $$\alpha $$ -optimistic value. Further, an $$\alpha $$ -optimal model for the uncertain random bottleneck assignment problem under the Hurwicz criterion is presented. Finally, a numerical example is given to illustrate how to use the proposed simulation algorithm to solve an uncertain random bottleneck assignment problem.

Online Energy Management and Heterogeneous Task Scheduling for Smart Communities with Residential Cogeneration and Renewable Energy

With the development of renewable energy technology and communication technology in recent years, many residents now utilize renewable energy devices in their residences with energy storage systems. We have full confidence in the promising prospects of sharing idle energy with others in a community. However, it is a great challenge to share residents’ energy with others in a community to minimize the total cost of all residents. In this paper, we study the problem of energy management and task scheduling for a community with renewable energy and residential cogeneration, such as residential combined heat and power system (resCHP) to pay the least electricity bill. We take elastic and inelastic load demands into account which are delay intolerant and delay tolerant tasks in the community. The minimum cost problem of a non-cooperative community is extracted into a random non-convex optimization problem with some physical constraints. Our objective is to minimize the time-average cost for each resident in the community, including the cost of the external grid and natural gas. The Lyapunov optimization theory and a primal-dual gradient method are adopted to tackle this problem, which needs no future data and has low computational complexity. Furthermore, we design a cooperative renewable energy sharing algorithm based on State-action-reward-state-action (Sarsa) Algorithm, in the condition that each residence in the community is able to communicate with its neighbors by a central controller. Finally, extensive simulations are presented to validate the proposed algorithms by using practical data.

Some Inequalities Combining Rough and Random Information.

Rough random theory, generally applied to statistics, decision-making, and so on, is an extension of rough set theory and probability theory, in which a rough random variable is described as a random variable taking “rough variable” values. In order to extend and enrich the research area of rough random theory, in this paper, the well-known probabilistic inequalities (Markov inequality, Chebyshev inequality, Holder’s inequality, Minkowski inequality and Jensen’s inequality) are proven for rough random variables, which gives a firm theoretical support to the further development of rough random theory. Besides, considering that the critical values always act as a vital tool in engineering, science and other application fields, some significant properties of the critical values of rough random variables involving the continuity and the monotonicity are investigated deeply to provide a novel analytical approach for dealing with the rough random optimization problems.