# An Adaptive Linear Programming Algorithm with Parameter Learning
## 1 Introduction
### An Example and Explanation Using KKT Conditions
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Weight**} & \multirow{2}{*}{**Starting Point**} & \multicolumn{2}{c}{**COBYLA**} & \multicolumn{2}{c}{**Trust-Constr**} & \multicolumn{2}{c}{**SLSOF**} & \multicolumn{2}{c}{**ALP**} & \multicolumn{2}{c}{**NSGA III/III–Population (P) = 20/50**} \\ \cline{3-10}  & & **Solution** & \(\sum_{k=1}^{2}w_{1}\) - f(v) & **Solution** & \(\sum_{k=1}^{2}w_{1}\) - f(v) & **Solution** & \(\sum_{k=1}^{2}w_{1}\) - f(v) & **Solution** & \(\sum_{k=1}^{2}w_{1}\) - f(v) & **Solution** & \(\sum_{k=1}^{2}w_{1}\) - f(v) \\ \hline \multirow{3}{*}{(1,0)} & (0,5,1) & & & & & P = 20: & P = 20: \\  & (0,0) & & & & & & G (0.55, 1.82) & 0.99 \\  & (2,0,5) & & & & & P = 50: & P = 50: \\ \cline{2-10}  & (0,5,1) & & & & & & A (0.51, 1.82) & 1 \\ \cline{2-10}  & (0,5,1) & & & & & P = 20: & P = 20: \\ \cline{2-10} \multirow{3}{*}{(0,1)} & (0,0) & & & & & P = 50: & P = 50: \\ \cline{2-10}  & (0,5,1) & & & & & P = 50: & P = 50: \\ \cline{2-10}  & (0,5,1) & & & & & & P = 50: \\ \cline{2-10}  & (0,5,1) & & & & & & P = 20: & P = 20: \\ \cline{2-10}  & (0,0,0) & & & & & & P = 50: & P = 50: \\ \cline{2-10}  & (0,5,1) & & & & & & D (0.51, 1.86) & 15.36 \\ \hline \multirow{3}{*}{(0,5,0.5)} & (0,0) & & & & & & P = 20: & P = 20: \\  & (0,0) & & & & & & P = 20: & P = 50: \\ \cline{2-10}  & (0,5,1) & & & & & & D (0.53, 1.87) & 7.78 \\ \cline{1-1} \cline{2-10}  & (0,5,1) & & & & &  
For a multi-objective (multi-goal) problem with nonconvex functions, when the scale of the objectives varies largely, optimizing algorithms cannot return feasible solutions, whereas a satisficing strategy may allow designers to identify adequate solutions. This is explained using the KKT conditions hereafter. When using optimizing algorithms to solve optimization problems, the first-order derivative of the Lagrange equation with respect to decision variable \(x\), which is a function of the parameters \(\mathcal{P}\) of the model (the coefficients in objectives and constraints), decision variables \(x\) (if any objective or constraint is nonlinear), Lagrange multipliers \(\mu\) and \(\lambda\), and weights \(\mathcal{P}\), combining the multiple goals, is shown in Equation (1). \[\
abla_{x}L(x,\mu,\lambda)=\mathcal{V}(\mathcal{P},x,\mu,\lambda,\,\mathcal{P}) \tag{1}\]  
For a satisficing strategy, the first-order derivative of the Lagrange equation contains only the coefficients of the deviation variables in the objective, since only deviation variables \(d^{\mp}\) constitute the objective (not the decision variables, \(x\)). For a \(\kappa\)-goal cDSP with \(m\) inequality constraints \(g(x)\) and \(\ell\) equality constraints \(h(x)\), if we use weights to combine the \(\kappa\) goals \(G(x,d)\), i.e., using the Archimedean strategy to manage a multi-goal cDSP, then the coefficients in the first-order Lagrange equation are only the weights and \(\tau\)--\(\tau\) is the Lagrange multiplier of the goal functions; see Equation (2). \[ \begin{split}&\
abla_{d}L(x^{s},d,\mu,\lambda,\tau)=\
abla_{d} \bar{z}(d)+\sum_{i=1}^{m}\mu_{i}\
abla_{d}g_{i}(x^{s})-\sum_{j=1}^{\ell} \lambda_{j}\
abla_{d}h_{j}(x^{s})-\sum_{k=1}^{\kappa}\tau_{k}\
abla_{d}G(x^{s},d)\\ &=\mathcal{V}(\mathcal{P},\tau)\end{split}  \tag{2}\]  
when using optimization, the second-order Lagrange equation may still have parameters and decision variables due to nonlinearity; see Equation (3). For satisficing, the second-order Lagrange equation with respect to deviation variables degenerates to zero because the objective of a cDSP is a linear combination of deviation variables; see Equation (4). That is why satisficing solutions do not need to meet the second-order KKT conditions.