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...) problems<span style="font-size: 8pt; position:relative; bottom: 0.3em;">[1]</span>. ECP can be thought as an extension of Kelley's cutting plane metho '''Step 0.''' Set k=1. Leave out all nonlinear constraints and set up an MILP problem. ...

...ng problems by using cutting planes in the late fifties and early sixties (1). \sum_{j}a_{i,j} x_j \leq b_i \qquad x_j \in \{0,1\} </math> ...

</math> （1） 1) The explicit ODE discretization holds little computational advantage becau ...

<math >~~~~~~~~~~y_{1},y_{2} \in \big\{0,1\big\}^{t} </math><br> ...play=inline>y^{1}</math>. Set the iteration counter <math display=inline>K=1</math>. Initialize the lower bound <math display=inline>Z_{L}=-\infty</math ...

...le f(x), \\ yf(x) + \tfrac{1}{2}c[f(x)]^2, & \text{if } -y/c \le f(x) \le (1-y)/c \\ -y^2/2c, & \text{if } f(x) \le -y/c \end{cases} </math> <br/> <math> 0 \le y \le 1, 0 < c </math> <br/> ...

...variability in the parameters of the problem at hand and or its solution [1]. In robust optimization, the modeler aims to find decisions that are optim '''1. Ignorance''' - such as not knowing exactly how much oil is in a reserve <b ...

...{1},x_{2} </math> and multiplying them by their weights to obtain <math>w_{1}, w_{2} </math>. All of the weights will be added up together to create the <math>w_{ij}(t+1) = w_{ij}(t) - \epsilon \frac{\partial E}{\partial w_{ij}}(t)</math> ...

...gative, but the coefficient <math>c</math> can only be positive<ref name=":1">S. Boyd, L. Vandenberghe, ''Convex Optimization''. Cambridge University Pr ...unction, or a posynomial, is defined as: <math>f(x)</math> = <math>\sum_{k=1}^K</math><math>c_kx_1^{a_{1k}}</math><math>x_2^{a_{2k}}</math>...<math>x_n^ ...

...<ref>Bradley, Stephen P. Applied Mathematical Programming. Addison-Wesley. 1 February 1977. 320-342. 18 Nov 2020</ref> Such a standard framework is used '''Step 1: Specify the stages of the problem''' <br /> ...

...iteration of Newton's method is usually written as: <math>x_{k+1}=x_k-H^{-1}\cdot\bigtriangledown f(x_k) </math>, where <math>k </math> is the iterati ...th>{df \over dx}=0, ||\bigtriangledown f(x)||<\epsilon \text{ or } |f(x_{k+1})-f(x_k)|<\epsilon </math> ...

<math>f_1(x) \le M\ast(1-y_1)</math> <math>f_2(x) \le M\ast(1-y_2)</math> ...

...Any given value <math>x'</math> between <math>x_{i}</math> and <math>x_{i+1}</math> can be written as the following form: <math>x'= \lambda x_{i}+(1-\lambda )x_{i+1}</math> (1) ...

...c programs.<span style="font-size: 8pt; position:relative; bottom: 0.3em;">1,2,3</span> :<math>s.t. ~~ f_i(x) \le 1, i = 1,...,m </math> ...

<math>L(x, \lambda ) = f(x) + \sum_{i=1}^{p}\lambda _{i}\cdot a_{i}(x)</math> <br> <math>\left [ \frac{\partial g(x)}{\partial x_{1}} ..... \frac{\partial g(x)}{\partial x_{n}}\right ]^{T} = \left [ 0 ... 0 ...

...l moving average of gradients and square gradients. And the parameters of β1 and β2 are used to control the decay rates of these moving averages. Adam '''''<math>w_t+1=w_t-update</math>''''' ...

...'Posynomial''' functions are defined as positive polynomials. <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1 s.t. & \quad f_i(x) \leq 1 \quad i = 1,....,m \\ ...

...on corresponding to every extreme point in the feasible region.<ref name=":1">Vanderbei, R. J. (2000). ''Linear programming: Foundations and extensions' ...l Engineering. Springer, New York, NY. <nowiki>https://doi.org/10.1007/978-1-4757-4106-3_8</nowiki></ref> ...

</math> non-linear programming (NLP) problems¹. Optimization of these non-convex functions f(x) is challenging since [[File:Image of updated estimators and envelopes.png|thumb|Figure 1: Relationships between the given function, concave over-estimators, convex ...

...70s. <ref name="two"> J. Rubin, [https://www.jstor.org/stable/25767684?seq=1 "A Technique for the Solution of Massive Set Covering Problems, with Applic <math> y_i = \begin{cases} 1, & \text{if subset }i\text{ is selected} \\ 0, & \text{otherwise } \end{cas ...

...t of actions and states available and the decision making frequency.<math>^1</math> This article will focus on discrete MDPs with finite states and fini ...lements: the agent, states, a model, actions, rewards, and a policy.<math>^1</math> The agent is the object or system being controlled that has to make ...