# An Adaptive Linear Programming Algorithm with Parameter Learning
## 1 Introduction
### An Example and Explanation Using KKT Conditions
\[\
abla_{xx}^{2}L(x,\mu,\lambda)=\
abla_{x}\mathcal{V}(\mathcal{P},x,\mu, \lambda,\mathcal{P})=\mathcal{V}^{\prime}(\mathcal{P}^{\prime},x) \tag{3}\]  
\[\
abla_{dd}^{2}L(x^{s},d,\mu,\lambda,\tau)=\
abla_{d}(\mathcal{V}(\mathcal{P},\tau))\equiv 0 \tag{4}\]  
Although both optimal solutions \(x^{s}\) and satisficing solutions \(x^{s}\) meet the first-order KKT conditions, the chance of maintaining the first-order KKT conditions for the two strategies under uncertainties varies. If any uncertainty with probability \(P\) takes place to an item \(\Im\) in the first-order equation that destroys its equilibrium, we denote it as \(\Pr\left(\widetilde{\Im}\left|P\right.\right)\). For an N-dimension, Q-parameter (here, we define a coefficient or an intercept of a constraint or an objective as a parameter. A parameter has a given value (either a constant value or a stochastic value) and the value does not depend on any decision variables), and \(\kappa\)-goal problem, using optimizing strategy, the source of \(\widetilde{\Im}\) can be decision variables \(\widetilde{x}_{n}\), Lagrange multipliers \(\widetilde{u}_{i}\) and \(\widetilde{\lambda}_{\gamma}\), and weights \(\widetilde{\mathcal{P}}_{k}\); for satisficing strategy, the source of \(\widetilde{\Im}\) can only be the weights \(\widetilde{\mathcal{P}}_{k}\) and the Lagrange multipliers for the goals \(\widetilde{\tau}_{k}\). If and only if none of the items under the uncertainty breaks the equilibrium of the first-order equation, then the optimal/satisficing solution is still optimal/satisficing under this uncertainty. For an N-dimension, Q-parameter, and \(\kappa\)-goal problem, the probabilities of maintaining an optimal solution and a satisficing solution under Uncertainty \(P\) are given in Equations (5) and (6), respectively. \[\Pr(x^{s}\left|P\right.)\approx\prod_{a=1}^{Q}\left[1-\Pr\left(\widetilde{ \mathcal{P}}_{a}\left|P\right.\right)\right]\prod_{n=1}^{N}[1-\Pr\left( \widetilde{x_{n}}\left|P\right.\right)]\prod_{i=1}^{m}[1-\Pr\left(\widetilde{u }_{i}\left|P\right.\right)]\prod_{j=1}^{\ell}\left[1-\Pr\left(\widetilde{ \lambda}_{j}\left|P\right.\right)\right]\prod_{k=1}^{\kappa}\left[1-\Pr\left( \widetilde{\mathcal{P}}_{k}\left|P\right.\right)\right] \tag{5}\]  
\[\Pr(x^{s}\left|P\right.)\approx\prod_{k=1}^{\kappa}\left[1-\Pr\left(\widetilde{ \mathcal{P}}_{k}\left|P\right.\right)\right]\prod_{k=1}^{\kappa}[1-\Pr\left( \widetilde{\tau}_{k}\left|P\right.\right)] \tag{6}\]  
As the value of any probability is in the range of \([0,1]\), the more items on the right-hand side we multiply (the more items the probability depends on), the lower the probability becomes. The items in Equation (6) are fewer than those of Equation (5). Hence, the chance of maintaining an optimal solution under Uncertainty P is often smaller than the chance of maintaining a satisficing solution with the same uncertainty; Equation (7). \[\Pr(\x^{*}\,|\P)\leq\Pr(\x^{*}\,|\P) \tag{7}\]  
In summary, using the satisficing strategy, we are less likely to lose a solution due to uncertainty for nonconvex problems with multiple objectives that have various scales; see Equations (8) and (9). Using satisficing, designers can deal with nonconvex, multi-objective problems that may be incomplete or inaccurate and with uncertainties, which helps remove Assumptions 1 and 2. \[\left[\Pr\left(\mathcal{S}\
eq\varnothing\middle|\left\{ \begin{array}{c} \text{Nonconvexity}\\ \text{objectives with various scales}\end{array} \right\}\right)\right]_{\text{ Satisficing}}\geq\]  
\[\left[\Pr\left(\mathcal{S}\
eq\varnothing\middle|\left\{ \begin{array}{c} \text{Nonconvexity}\\ \text{objectives with various scales}\end{array} \right\}\right)\right]_{\text{ Optimizing}} \tag{8}\]  
\[\left[\Pr\left(\x^{*}\middle|\P\right)\right]_{\text{Satisficing}}\geq\left[ \Pr\left(\x^{*}\middle|\P\right)\right]_{\text{Optimizing}} \tag{9}\]