Linear Superiorization for Infeasible Linear Programming

Linear superiorization (abbreviated: LinSup) considers linear programming (LP) problems wherein the constraints as well as the objective function are linear. It allows to steer the iterates of a feasibility-seeking iterative process toward feasible points that have lower (not necessarily minimal) values of the objective function than points that would have been reached by the same feasiblity-seeking iterative process without superiorization. Using a feasibility-seeking iterative process that converges even if the linear feasible set is empty, LinSup generates an iterative sequence that converges to a point that minimizes a proximity function which measures the linear constraints violation. In addition, due to LinSup's repeated objective function reduction steps such a point will most probably have a reduced objective function value. We present an exploratory experimental result that illustrates the behavior of LinSup on an infeasible LP problem.Comment: arXiv admin note: substantial text overlap with arXiv:1612.0653

Linear Programming the Applications in Agriculture

‘Linear programming\u27 adalah teknik permodelan matematika yang didisain untuk mengop-timastkan penggunaan sumber-sumber yang teratas; sifatnya deterministik yaitu dalam kondsi infarmasi data yang lengkap. Sebagai salah satu dari berbagai metoda kuantftatif dalam masalah optimasi, dapat diaplikasikan tidak hanya dalam sektor industri tetapi iuga dalam sektor pertanian. Model matematika ‘linear programming\u27 bagi masalah pertanian yang diambil dari contah kasus di Tanzania, variabelnya adalah luas area yang ditanami, batasan-batasannya seperti ketersediaan lahan, ketersediaan tenaga kerja, kebutuhan akan makanan, dan batasan non-negatif. Fungsi tuiuannya, merupakan salah safu dari beberapa kemungkinan fungsi tuiuan, yaitu memaksimumkan penerimaan tahunan bersih

Aplikasi Fuzzy Linear Programming untuk Produksi Bola Lampu di PT XYZ

PT XYZ merupakan Perusahaan yang memproduksi bola lampu. Permintaan pasar yang tinggi menyebabkan Perusahaan tidak dapat memenuhi permintaan tersebut dikarenakan perencanaan produksi yang tidak optimal. Dari data, diketahui bahwa Perusahaan tidak dapat memenuhi permintaan pasar pada produk bola lampu merek Stanlee Star G-20 sebesar 8% dan S-25 sebesar 18,32% sedangkan bola lampu merek Dai-Ichi G40 diproduksi melebihi permintaan pasar sebesar 9,1%. Hal ini menyebabkan Perusahaan kehilangan opportunity profit. Perencanaan produksi bola lampu diteliti dengan tujuan agar Perusahaan dapat memenuhi permintaan pasar sesuai dengan keterbatasan sumber daya yang tersedia. Metode perencanaan produksi yang digunakan adalah metode fuzzy linear programming dengan metode simpleks. Dengan menggunakan Fuzzy Linear Programming dapat diperoleh nilai optimum jumlah produk bola lampu yang diproduksi sesuai permintaan pasar dan sesuai dengan keterbatasan sumber daya produksi. Sumber daya yang diteliti adalah kapasitas produksi, waktu kerja, dan bahan baku. Nilai interval logika fuzzy yang digunakan adalah t = 0 dan t = 1. Penyelesaian metode simpleks dilakukan dengan menggunakan software LINGO 13. Hasil penilitian menunjukkan bahwa permintaan pasar terpenuhi untuk ketiga merek bola lampu. Kapasitas produksi mencukupi sehingga tidak diperlukan penambahan jumlah mesin, sedangkan waktu kerja dan bahan baku tidak mencukupi. Perusahaan dapat menentukan jumlah bahan baku dan waktu kerja yang diperlukan dengan menggunakan nilai λ yaitu sebesar 0,536. Nilai λ digunakan untuk menentukan skala terbesar nilai interval t untuk setiap kendala bahan baku dan waktu kerja yaitu 0,464. Aplikasi fuzzy linear programming meningkatkan keuntungan sebesar 7,39% dari konsep linear programming biasa

Decoding by Linear Programming

This paper considers the classical error correcting problem which is frequently discussed in coding theory. We wish to recover an input vector $f \in \R^n$ from corrupted measurements $y = A f + e$. Here, $A$ is an $m$ by $n$ (coding) matrix and $e$ is an arbitrary and unknown vector of errors. Is it possible to recover $f$ exactly from the data $y$? We prove that under suitable conditions on the coding matrix $A$, the input $f$ is the unique solution to the $\ell_1$-minimization problem ($\|x\|_{\ell_1} := \sum_i |x_i|$) $$\min_{g \in \R^n} \| y - Ag \|_{\ell_1}$$ provided that the support of the vector of errors is not too large, $\|e\|_{\ell_0} := |\{i : e_i \neq 0\}| \le \rho \cdot m$ for some $\rho > 0$. In short, $f$ can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program). In addition, numerical experiments suggest that this recovery procedure works unreasonably well; $f$ is recovered exactly even in situations where a significant fraction of the output is corrupted.Comment: 22 pages, 4 figures, submitte

Robust distributed linear programming

This paper presents a robust, distributed algorithm to solve general linear programs. The algorithm design builds on the characterization of the solutions of the linear program as saddle points of a modified Lagrangian function. We show that the resulting continuous-time saddle-point algorithm is provably correct but, in general, not distributed because of a global parameter associated with the nonsmooth exact penalty function employed to encode the inequality constraints of the linear program. This motivates the design of a discontinuous saddle-point dynamics that, while enjoying the same convergence guarantees, is fully distributed and scalable with the dimension of the solution vector. We also characterize the robustness against disturbances and link failures of the proposed dynamics. Specifically, we show that it is integral-input-to-state stable but not input-to-state stable. The latter fact is a consequence of a more general result, that we also establish, which states that no algorithmic solution for linear programming is input-to-state stable when uncertainty in the problem data affects the dynamics as a disturbance. Our results allow us to establish the resilience of the proposed distributed dynamics to disturbances of finite variation and recurrently disconnected communication among the agents. Simulations in an optimal control application illustrate the results

A linear programming manual

Computer solutions of linear programming problems are outlined. Information covers vector spaces, convex sets, and matrix algebra elements for solving simultaneous linear equations. Dual problems, reduced cost analysis, ranges, and error analysis are illustrated