linear program

Linear programming is a mathematical optimization technique used to find the best possible solution to a problem with linear constraints. The goal of linear programming is to maximize or minimize a linear objective function subject to a set of linear constraints. The variables in the objective function and constraints are all linear, meaning they are first degree polynomials. Linear programming has many practical applications, such as in resource allocation, transportation planning, and production scheduling. The most common method for solving linear programming problems is the simplex method.

相关问题

We consider a persuasion problem between a sender and a receiver where utility may be nonlinear in the latter’s belief; we call such receivers risk conscious. Such utility models arise when the receiver exhibits systematic biases away from expected utility maximization, such as uncertainty aversion (e.g., from sensitivity to the variance of the waiting time for a service). Because of this nonlinearity, the standard approach to finding the optimal persuasion mechanism using revelation principle fails. To overcome this difficulty, we use the underlying geometry of the problem to develop a convex optimization framework to find the optimal persuasion mechanism. We define the notion of full persuasion and use our framework to characterize conditions under which full persuasion can be achieved. We use our approach to study binary persuasion, where the receiver has two actions and the sender strictly prefers one of them at every state. Under a convexity assumption, we show that the binary persuasion problem reduces to a linear program and establish a canonical set of signals where each signal either reveals the state or induces in the receiver uncertainty between two states. Finally, we discuss the broader applicability of our methods to more general contexts, and we illustrate our methodology by studying information sharing of waiting times in service systems翻译一下

我们考虑一个发件人和接收者之间的说服问题，其中接收者的效用可能在其信念中是非线性的；我们把这样的接收者称为风险感知者。这种效用模型出现在接收者偏离预期效用最大化的系统性偏差中，例如不确定性规避（例如，对于等待服务的等待时间方差的敏感性）。由于这种非线性，使用启示原则寻找最优说服机制的标准方法失败了。为了克服这个难点，我们利用问题的基本几何形状，开发了一个凸优化框架来找到最优的说服机制。我们定义了完全说服的概念，并使用我们的框架来表征实现完全说服的条件。我们使用我们的方法研究了二元说服问题，其中接收者有两个行动，发件人在每个状态下都严格偏好其中一个。在凸性假设下，我们证明了二元说服问题可以简化为线性规划，并建立了一个规范信号集，其中每个信号要么揭示状态，要么在接收者两个状态之间引起不确定性。最后，我们讨论了我们的方法对更一般情境的广泛适用性，并通过研究服务系统中等待时间的信息共享来说明我们的方法论。

(Geometry: intersecting point) Suppose two line segments intersect. The two end-points for the first line segment are (x1, y1) and (x2, y2) and for the second line segment are (x3, y3) and (x4, y4).Write a program that prompts the user to enter these four endpoints and displays the intersecting point. the intersecting point can be found by solving a linear equation. Write the LinearEquation class in Programming to solve this equation. input style : Enter eight double number output style: Displays the intersecting point or The two lines are parallel

Here's one possible solution in Java:

import java.util.Scanner;

public class IntersectingPoint {
    public static void main(String[] args) {
        // Prompt the user to enter the four endpoints
        Scanner input = new Scanner(System.in);
        System.out.print("Enter x1, y1, x2, y2, x3, y3, x4, y4: ");
        double x1 = input.nextDouble();
        double y1 = input.nextDouble();
        double x2 = input.nextDouble();
        double y2 = input.nextDouble();
        double x3 = input.nextDouble();
        double y3 = input.nextDouble();
        double x4 = input.nextDouble();
        double y4 = input.nextDouble();
        
        // Create two LinearEquation objects for the two line segments
        LinearEquation eq1 = new LinearEquation(y1 - y2, x2 - x1, y1*x2 - y2*x1);
        LinearEquation eq2 = new LinearEquation(y3 - y4, x4 - x3, y3*x4 - y4*x3);
        
        // Check if the two lines are parallel
        if (eq1.isParallel(eq2)) {
            System.out.println("The two lines are parallel");
        } else {
            // Compute the intersecting point
            double x = eq1.getX(eq2);
            double y = eq1.getY(eq2);
            System.out.printf("The intersecting point is (%.2f, %.2f)\n", x, y);
        }
    }
}

class LinearEquation {
    private double a, b, c;

    public LinearEquation(double a, double b, double c) {
        this.a = a;
        this.b = b;
        this.c = c;
    }

    public double getA() {
        return a;
    }

    public double getB() {
        return b;
    }

    public double getC() {
        return c;
    }

    public boolean isParallel(LinearEquation other) {
        return Math.abs(a * other.b - b * other.a) < 1e-6;
    }

    public double getX(LinearEquation other) {
        return (c * other.b - b * other.c) / (a * other.b - b * other.a);
    }

    public double getY(LinearEquation other) {
        return (a * other.c - c * other.a) / (a * other.b - b * other.a);
    }
}

The program first prompts the user to enter the four endpoints, then creates two `LinearEquation` objects for the two line segments, and finally checks if the two lines are parallel or computes the intersecting point using the `getX` and `getY` methods of the `LinearEquation` class. The `isParallel` method checks if the two lines are parallel by comparing the slopes of the two lines, and the `getX` and `getY` methods solve the system of two linear equations for the intersecting point.