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$$ \text{Maximize } f(\boldsymbol{x}) = \sum_{i=1}^n x_i$$ $$x_i \in {0,1}$$ $$f(\boldsymbol{x}^*) = f(1,1,\dots,1) = n$$
$$ \text{Maximize } f(\boldsymbol{x}) = \sum_{i=1}^n F(x_i)$$ $$F(x_i) = \cases{1 & \text{if } x_i=1\cr 0 & \text{otherwise}}$$ $$x_i \in {0,1,\dots,5}$$ $$f(\boldsymbol{x^*}) = f(1,1,\dots,1) = n$$
Given a set of $$n$$ items, each with a value $$v_i$$ and a cost $$w_i$$ , along with a maximum capacity $$W(=600)$$
$$ \text{Maximize } f(\boldsymbol{x}) = \sum_{i=1}^n v_ix_i$$ $$\text{Subject to } c(\boldsymbol{x}) = \sum_{i=1}^n w_ix_i \leq W$$ $$x_i \in {0,1}$$
Multiple Knapsack problem: Given a set of $$n$$ items, each with a value $$v_i$$ and a cost $$w_i$$ , along with maximum capacitys $$\boldsymbol{W}(={500,300,100})$$
$$ \text{Maximize } f(\boldsymbol{x}) = \sum_{j=1}^3\sum_{i=1}^n v_iF(x_i,j)$$ $$F(x_i,j) = \cases{1 & \text{if } x_i=j\cr 0 & \text{otherwise}}$$ $$\text{Subject to } c_j(\boldsymbol{x}) = \sum_{i=1}^n w_iF(x_i,j) \leq W_j$$ $$x_i \in {0,1,2,3}$$
$$ \text{Minimize } f(\boldsymbol{x}) = -20 \exp(-0.2 \sqrt{\frac{1}{n} \sum_{i=1}^n x_i^2}) - \exp(\frac{1}{n} \sum_{i=1}^n \cos(2\pi x_i)) + 20 + \exp(1)$$
$$-5 \leq x_i \leq 5$$ $$f(\boldsymbol{x}^*) = f(0,0,\dots,0)=0 $$
$$ \text{Minimize } f(\boldsymbol{x}) = \sum_{i=1}^{n-1} (100(x_{i+1}-x_i^2)^2+(1-x_i)^2)$$
$$-5 \leq x_i \leq 5$$ $$f(\boldsymbol{x}^*) = f(1,1,\dots,1)=0 $$
$$ \text{Minimize } f(\boldsymbol{x}) = \sum_{i=1}^n x_i^2$$
$$-5 \leq x_i \leq 5$$ $$f(\boldsymbol{x}^*) = f(0,0,\dots,0)=0 $$
Weight Vector Generation Method You can change the weight vector size in the animation
Weight Vector Change Methods You can change the weight vector distribution in the animation
