Disjunctive Aspects in Generalized Semi-infinite Programming

The feasible set $M$ in general semi-infinite programming (GSIP) need not be closed. This fact is well known. We introduce a natural constraint qualification, called symmetric Mangasarian-Fromovitz constraint qualification (Sym-MFCQ). The Sym-MFCQ is a nontrivial extension of the well-known (extended) MFCQ for the special case of semi-infinite programming (SIP) and disjunctive programming. Under the Sym-MFCQ the closure $\overline{M}$ has an easy and also natural description. As a consequence, we get a description of the interior and boundary of $M$. The Sym-MFCQ is shown to be generic and stable under $C^1$-perturbations of the defining functions. For the latter stability the consideration of the closure of $M$ is essential. We introduce an appropriate notion of Karush-Kuhn-Tucker (KKT) points. We show that local minimizers are KKT points under the Sym-MFCQ.

A generalized benders decomposition-based global optimization approach to symbolic regression for explicit surrogate modeling from limited data information

In this paper, our objective is to develop a mathematical formulation of solving the Bin Packing Problem (BPP) with different sizes of box. It has become one of the most important mathematical applications throughout the time. In our study, Modified Branch and Bound Algorithm (MBBA) is developed to generate all the feasible packing patterns of given boxes to required containers for One-Dimensional BPP. Further development of algorithms was made to ascertain the locations of each box within the containers by using Cartesian coordinate system. Developed algorithms are coded and programmed in the Python programming environment to generate feasible packing patterns.

Reprint of: Enhancing natural gas-to-liquids (GTL) processes through chemical looping for syngas production: Process synthesis and global optimization

The number of real-life applications for standard and general semi-infinite programming is vast.KeywordsMinimax ProblemChebyshev ApproximationTool Center PointDisjunctive ProgrammingRobust Optimization ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A First-Order Optimality Condition in Nonsmooth Generalized Semi-infinite Programming (GSIP)

Recently, conjunctive use of surface water and ground water has been one of the \nefficient ways in optimal management of water resources. Using conjunctive optimal \nuse and management of conjunctive, we can compensate the nonuniform spatial and \ntemporal distribution of precipitation. Without recent advances in computer \nhardware and software technology such as geographic information system (GIS) and \ndata base management system technology (DBMS), it is not possible to develop a \ncomputer base conjunctive management of groundwater and surface water model. \nDeveloped Network flow models for analyzing optimal management of conjunctive use \nof surface water and groundwater have been used. The real physical system is \nrepresented in this system as a network system including nodes and links. Nodes are \nused for representing storage and non-storage points and links represent the \nnatural channels or canals as connecting elements. Dynamically designing of this \nmodel allows the user to interactively develop the given water resource system. A \nmixed integer linear programming is used for optimum conjunctive operation of \ngroundwater and surface water systems. In this study, conjunctive surface and \nground water resources model has been applied in Tehran metropolitan. Tehran field \nwater resources, which meet the potable, agricultural, industrial, environmental \nand urban water demands, include five reservoirs and several aquifers. Results \nobtained show the ability of the tools and the methods in integrated management of \nlarge scale water resources systems.

This paper shows that any linear disjunctive program with a finite number of constraints can be transformed into an equivalent facial program. Based upon linear programming technique, a new, finite cutting plane method is presented for the facial programs.

An analysis is performed of many standard and linearized electrooptic modulators known in the industry. The transfer functions of these modulators are evaluated under a consistent set of performance figures of merit, which are gain and spur-free dynamic range, using a canonical set of optical link parameters. The tolerance of the needed precision of the parameters of the linearization mechanisms of all of these modulators is compared over the entire interesting range of noise bandwidth. A computer program was written to analyze the frequency dependence of any modulator transfer function under any set of functional inputs. The program is used to illustrate and compare the frequency dependence of the figures of merit of all of the modulators for which a d-c analysis was performed. Further analysis looks at the effect of greater noise-bandwidth and recovering the frequency-dependent degradation of gain and dynamic range through re-phasing techniques. The gain of directional couplers is analyzed in-depth. Two novel modulator schemes are produced. The first uses reflective wave techniques to retime the electrical and optical waves half way through the modulator. The second uses fabrication geometry and properties of the linearization technique to make a more robust modulator (applicable to three of the modulators analyzed). A 94 GHz antenna-coupled directional coupler modulator was initially demonstrated using an old modulator chip from Finbar Sheehy. A peculiar bug with the chip was uncovered. And a new modulator experiment was constructed and many aspects of the experimental apparatus were optimized. Though the revised experiment ultimately did not yield modulation side bands, it did couple a 94 GHz microwave signal into the optical waveguide, and many interesting challenges of high frequency electrooptic modulator fabrication were evaluated and improved upon.

Solving optimization problems with sparse or low-rank optimal solutions has been an important topic since the recent emergence of compressed sensing and its matrix extensions such as the matrix rank minimization and robust principal component analysis problems. Compressed sensing enables one to recover a signal or image with fewer observations than the “length” of the signal or image, and thus provides potential breakthroughs in applications where data acquisition is costly. However, the potential impact of compressed sensing cannot be realized without efficient optimization algorithms that can handle extremely large-scale and dense data from real applications. Although the convex relaxations of these problems can be reformulated as either linear programming, second-order cone programming or semidefinite programming problems, the standard methods for solving these relaxations are not applicable because the problems are usually of huge size and contain dense data. In this dissertation, we give efficient algorithms for solving these “sparse” optimization problems and analyze the convergence and iteration complexity properties of these algorithms. Chapter 2 presents algorithms for solving the linearly constrained matrix rank minimization problem. The tightest convex relaxation of this problem is the linearly constrained nuclear norm minimization. Although the latter can be cast and solved as a semidefinite programming problem, such an approach is computationally expensive when the matrices are large. In Chapter 2, we propose fixed-point and Bregman iterative algorithms for solving the nuclear norm minimization problem and prove convergence of the first of these algorithms. By using a homotopy approach together with an approximate singular value decomposition procedure, we get a very fast, robust and powerful algorithm, which we call FPCA (Fixed Point Continuation with Approximate SVD), that can solve very large matrix rank minimization problems. Our numerical results on randomly generated and real matrix completion problems demonstrate that this algorithm is much faster and provides much better recoverability than semidefinite programming solvers such as SDPT3. For example, our algorithm can recover 1000 × 1000 matrices of rank 50 with a relative error of 10−5 in about 3 minutes by sampling only 20 percent of the elements. We know of no other method that achieves as good recoverability. Numerical experiments on online recommendation, DNA microarray data set and image inpainting problems demonstrate the effectiveness of our algorithms. In Chapter 3, we study the convergence/recoverability properties of the fixed-point continuation algorithm and its variants for matrix rank minimization. Heuristics for determining the rank of the matrix when its true rank is not known are also proposed. Some of these algorithms are closely related to greedy algorithms in compressed sensing. Numerical results for these algorithms for solving linearly constrained matrix rank minimization problems are reported. Chapters 4 and 5 considers alternating direction type methods for solving composite convex optimization problems. We present in Chapter 4 alternating linearization algorithms that are based on an alternating direction augmented Lagrangian approach for minimizing the sum of two convex functions. Our basic methods require at most O(1/e) iterations to obtain an e-optimal solution, while our accelerated (i.e., fast) versions require at most O(1/ 3 ) iterations, with little change in the computational effort required at each iteration. For more general problem, i.e., minimizing the sum of K convex functions, we propose multiple-splitting algorithms for solving them. We propose both basic and accelerated algorithms with O(1/e) and O(1/ 3 ) iteration complexity bounds for obtaining an e-optimal solution. To the best of our knowledge, the complexity results presented in these two chapters are the first ones of this type that have been given for splitting and alternating direction type methods. Numerical results on various applications in sparse and low-rank optimization, including compressed sensing, matrix completion, image deblurring, robust principal component analysis, are reported to demonstrate the efficiency of our methods.

This paper focuses on solving two-stage stochastic mixed integer programs (SMIPs) with general mixed integer decision variables in both stages. We develop a decomposition algorithm in which the first stage approximation is solved using a branch-and-bound tree with nodes inheriting Benders' cuts that are valid for their ancestor nodes. In addition, we develop two closely related convexification schemes which use multi-term disjunctive cuts to obtain approximations of the second stage mixed-integer programs. We prove that the proposed methods are finitely convergent. One of the main advantages of our decomposition scheme is that we use a Benders-based branch-and-cut approach in which linear programming approximations are strengthened sequentially. Moreover as in many decomposition schemes, these subproblems can be solved in parallel. We also illustrate these algorithms using several variants of an SMIP example from the literature, and present preliminary evidence of the scalability of these algorithms as the number of scenarios increases.

Circular programming problems are a new class of convex optimization problems that include second-order cone programming problems as a special case‎. ‎Alizadeh and Goldfarb [Math‎. ‎Program‎. ‎Ser‎. ‎A 95 (2003) 3--51] introduced primal-dual path-following algorithms for solving second-order cone programming problems‎. ‎In this paper‎, ‎we generalize their work by using the machinery of Euclidean Jordan algebras associated with the circular cones to derive primal-dual path-following interior point algorithms for circular programming problems‎. ‎We prove polynomial convergence of the proposed algorithms by showing that the circular logarithmic barrier is a strongly self-concordant barrier‎. ‎The numerical examples show the path-following algorithms are simple and efficient‎.

Network simulation is a technique that models network behavior by performing transaction calculations between different network entities and using mathematical formulas and taking observations of network products. A network simulator is a software program have been applied to analyze the performance of a computer network without the presence of a real network. Hardware equipment, equipment configuration, communication, and routing protocols and network traffic modeled in simulation software and the behavior of the network and its components examined from different dimensions. The user can also customize the simulation software according to their needs. Simulation software has different uses, and the user can use these tools to model their network by recognizing this software. In terms of research, it is difficult to create a network, especially large networks, in a real-time scenario, and it is not easily possible to carry out it in the real world, and it is very costly. So, simulators help network developers to control whether the network can work in real-time or not, or whether it is efficient enough. This reduces the time and cost of network application testing.Today, simulation technology is successfully used to model, design and manage a variety of intelligent systems. Numerous tools have been created in this regard. In this article, we review and compare important network simulators such as CloudSim, GloMoSim, GNS3, NS-2, Opnet, OMNet ++, NetSim, NS-3, AVRORA, Packet Tracer, QualNet, J-Sim, REAL and OptSim and their results. These comparisons express from several perspectives in the tables.

Accurately representing and understanding the dynamics driving the global carbon cycle are of strong significance for the study of the Earth System as well as for reliable climate change projections. Model development in the biogeochemistry field traditionally relies on empirical studies and on already established theoretical foundations. With increased data availability, model development in the field of biogeochemistry has started to open more to the use of machine learning approaches for helping to validate and calibrate the existing model formulations. However, the validity of the studied model structures are not often debated. This thesis introduces a novel framework for modelling biogeochemistry fluxes by using symbolic regression approaches to automatically generate interpretable mathematical models. The thesis starts by first illustrating the potential of gene expression programming (GEP) to discover interesting models as mathematical formulas based entirely on real time series data measured at a single monitoring site. The GEP discovered models perform better predictions than already established models in the ecology community. Further, the GEP models have the advantage of being represented as mathematical formulas that can be used similarly to natural laws from the ecology community. Still, the complexity of GEP models makes it difficult to really interpret the described model dynamics. To tackle model complexity GEP is extended with CMA-ES for performing local parameter optimizations in the evolution process. The resulting algorithm is CMAGEP, a novel system that is a GEP and ES hybrid approach capable of delivering more accurate and more compact solutions compared to standard GEP. Generating compact solutions means that CMAGEP discovers mathematical models that can be more easily interpretable, and that can be more easily combined with already established knowledge. CMAGEP is successfully used for modelling various carbon fluxes; first it helps discover non-linear dynamics in the carbon cycle at an Arctic site and produce a very compact solution, and secondly, it reveals interesting and relevant patterns in the underlying processes determining the global terrestrial carbon exchanges. Considering the important results shown in this extensive interdisciplinary study it becomes clear that by introducing the new CMAGEP system, an important contribution was made to the field of symbolic regression by giving deserved attention to the often neglected aspect of interpretability. Furthermore, the application of CMAGEP in a symbolic regression framework to model terrestrial carbon fluxes helped build novel knowledge in the ecology field, giving this approach a significant potential for other future applications.

In the range emptiness searching problem, we are given a set P of n points in Rd, and wish to preprocess them into a data structure that supports efficient range emptiness queries, in which we specify a range σ, which is a semi-algebraic set in Rd of some fixed kind, and wish to determine whether P ∩ σ = θ. Range emptiness searching arises in many applications, and has been treated by Matousek [15] in the special case where the ranges are halfspaces bounded by hyperplanes. In this paper we extend the analysis to the general semi-algebraic case, and show how to adapt Matousek's technique to this case, without the need to linearize the ranges into a higher-dimensional space. This yields more efficient solutions to several interesting problems, and we demonstrate the new technique in two applications:(i) An algorithm for ray shooting amid balls in R3, which uses O* (n) storage and preprocessing, and answers a query in O* (n2/3) time, improving the previous bound of O* (n3/47).(ii) An algorithm that preprocesses, in O*(n) time, a set P of n points in R3 into a data structure with O*(n) storage, so that, for any query line l or segment e, the point of P farthest from l or from e can be computed in O* (n1/2) time.Our technique is closely related to the notions of nearest- or farthest-neighbor generalized Voronoi diagrams, and of the union or intersection of geometric objects, where sharper bounds on the combinatorial complexity of these structures yield faster range emptiness searching algorithms. For example, in the case of ray shooting amid balls, the structure that arises in our algorithm is the Euclidean Voronoi diagram of lines in 3-space, and the performance of the algorithm depends on the complexity of such diagrams (for which tight bounds are still unknown).