In Many-objective Optimization Problems (MaOP), the Pareto dominance has exponential increase of non-dominated solutions and the decrease of selection pressure with increasing optimization objectives. To solve these issues, a new type of dominance, namely Linear Weighted Minimal/Maximal dominance (LWM-dominance) was proposed based on the ideas of comparing multi-objective solutions by using linear weighted aggregation and Pareto dominance. It is theoretically proved that LWM non-dominated solution set is a subset of Pareto non-dominated solution set, meanwhile the important corner solutions are reserved. Furthermore, an MaOP algorithm based on LWM dominance was presented. The empirical studies proved the corollaries of the proposed LWM dominance. In detail, the experimental results in random objective space show that the LWM dominance is suitable for the MaOPs with 5-15 objectives; the experiment on comparing the number of LWM non-dominated solutions and Pareto non-dominated solutions with subjects of DTLZ1-DTLZ7 shows that the proportion of non-dominated solutions decreases by about 17% on average when the number of optimization objectives is 10 and 15.