# An Adaptive Linear Programming Algorithm with Parameter Learning
## 1 Introduction
### Mechanisms to Ensure That the cDSP and ALP Find Satisficing Solutions
The elements of such a nonnegative vector are Lagrange multipliers, \(\mu\) and \(\lambda\). The format of an optimization problem \(\mathbb{O}\):  
Given  
\[ \begin{array}{l}\mathrm{f}:\ \mathbb{R}^{n}\to\mathbb{R},\mathcal{F}\subseteq \mathbb{R}^{n}\\ \mathcal{F}=\left\{\mathrm{x}\in\mathbb{R}^{n}|\mathrm{g}_{i}(\mathrm{x})\geq 0,\ i=1,\ldots,\mathrm{m},\mathrm{h}_{j}(\mathrm{x})=0,\ j=1,\ldots,\ell\right\} \end{array} \]  
find  
\[\mathrm{x}^{*}:\ \mathrm{f}(\mathrm{x}^{*})\succcurlyeq\mathrm{f}(\mathrm{x}),\ \forall\mathrm{x}\in\mathcal{F}\]  
One variant of optimization is goal programming. The format of a goal programming problem \(\mathbb{O}^{\mathrm{goal}}\) is represented as follows. A target value \(\mathrm{T}\) is predefined for the objective function \(\mathrm{f}(\mathrm{x})\) as the right-hand side value, so the objective becomes an equation, and we call it a goal. \(\mathrm{d}^{-}\) and \(\mathrm{d}^{+}\) are deviation variables measuring the underachievement and overachievement of the goal towards its target. The problem is solved by minimizing the deviation variables, which is minimizing the difference between \(\mathrm{f}(\mathrm{x})\) and \(\mathrm{T}\). In other words, goal programming is aimed at finding \(\mathrm{T}\)'s closest projection on \(\mathcal{F}\). The format of a goal programming problem \(\mathbb{O}^{\mathrm{goal}}\):  
Given  
\[ \begin{array}{l}\mathrm{f}:\ \mathbb{R}^{n}\to\mathbb{R},\mathcal{F}\subseteq \mathbb{R}^{n}\\ \mathcal{F}=\left\{\mathrm{x}\in\mathbb{R}^{n}|\mathrm{g}_{i}(\mathrm{x})\geq 0,\ i=1,\ldots,\mathrm{m},\mathrm{h}_{j}(\mathrm{x})=0,\ j=1,\ldots,\ell,\mathrm{ d}^{-}\cdot\mathrm{d}^{+}=0,\ 0\leq\mathrm{d}^{\mp}\leq 1,\mathrm{Goal}:\ \mathrm{f}( \mathrm{x})+\mathrm{d}^{-}-\mathrm{d}^{+}=\mathrm{T}\right\}\\ \end{array} \]  
find  
\[\mathrm{x}^{*}:\mathbb{P}_{\mathrm{x}\in\mathcal{F}}(\mathrm{Goal}:\ \mathrm{f}( \mathrm{x})=\mathrm{T})\]  
In the cDSP, elements of mathematical programming and goal programming are combined. A cDSP \(\mathbb{C}\) is represented as follows. For a nonlinear cDSP, we first linearize the nonlinear equations, including nonlinear constraints and nonlinear goal. Therefore, the nonlinear cDSP first becomes a linear problem with a linear goal \(\mathrm{Goal}^{\mathrm{linear}}\), a linear feasible space \(\mathcal{F}^{\mathrm{li}}\) bounded by linear constraints \(\mathrm{g}(\mathrm{x})^{\mathrm{li}}\geq 0\) and \(\mathrm{h}(\mathrm{x})^{\mathrm{li}}=0\). Thus, using a cDSP, we seek the closest projection from the linear goal set onto a linear feasible set. We define the solution as a satisficing solution, and we use \(\mathrm{x}^{\mathrm{s}}\) to denote it. The format of a cDSP \(\mathbb{C}\):  
Given  
\[ \begin{array}{l}\mathrm{f}:\ \mathbb{R}^{n}\to\mathbb{R},\mathcal{F}\subseteq \mathbb{R}^{n}\\ \mathcal{F}^{\mathrm{li}}=\left\{\mathrm{x}\in\mathbb{R}^{n}|\mathrm{g}_{i}( \mathrm{x})^{\mathrm{li}}\geq 0,\ i=1,\ldots,\mathrm{m},\mathrm{h}_{j}(\mathrm{x})^{ \mathrm{li}}=0,\ j=1,\ldots,\ell,\mathrm{d}^{-}\cdot\mathrm{d}^{+}=0,\ 0\leq\mathrm{d}^{\mp}\leq 1, \mathrm{Goal}^{\mathrm{li}}:\frac{\mathrm{f}_{j}(\mathrm{x})^{\mathrm{li}}}{ \mathrm{T}}+\mathrm{d}^{-}-\mathrm{d}^{+}=1\right\}\\ \end{array} \]  
find  
\[\mathrm{x}^{*}:\mathbb{P}_{\mathrm{x}\in\mathcal{F}^{\mathrm{li}}}\Big{(} \mathrm{Goal}^{\mathrm{li}}:\mathrm{f}(\mathrm{x})^{\mathrm{li}}=\mathrm{T} \Big{)}\]  
The difference between \(\mathrm{x}^{*}\) and \(\mathrm{x}^{\mathrm{s}}\) is that \(\mathrm{x}^{*}\) conforms to both the necessary (first-order) and sufficient (second-order) KKT conditions, whereas \(\mathrm{x}^{\mathrm{s}}\) conforms to the necessary KKT condition but may not conform to the sufficient KKT condition.