Whenever I see “problem solving” listed as a component of emotional intelligence, I tend to regard the rest of what the author says on that subject with skepticism. Typically problem solving is considered a combination of creativity and logic to generate an innovative idea and put it to practical use. Creativity hardly seems like it is related to emotional intelligence, judging by the character of creative people. In fact researchers have found that one recurring trait of this group is a lack of regard for social skills. They tend to be less considerate, more likely to find fault with others, less agreeable and more rebellious than the average person. 
However, research is keeps popping up which shows a connection between emotional intelligence and problem solving. Some of these studies deal with problem solving by teams in which correlation is found between teams with higher totals on tests of emotional intelligence and their ability to complete problem solving tasks. It makes sense that team members who stubbornly try to hog all the attention or refuse to cooperate are going to impede the work of others. But there is also research that indicates that emotional intelligence gives individuals the edge in problem solving. 
The emotional facilitation of thinking and problem solving is included in Mayer, Salovey, and Caruso’s Multifactor Emotional Intelligence Scale Test (MEIS), a common tests used to assess emotional intelligence. This group of researches has conceived of “emotional intelligence” as a cognitive ability, and the test is composed of written and visual questions as opposed to putting individuals into situations in which observers rate their emotional intelligence. 
Research has shown that people who score high on both cognitive tests and the MEIS exhibit the best skills in problem solving. However, some researchers have found that higher cognitive skills are an even better predictor than high emotional intelligence for problem-solving skills, and other “life skills” such as lowering anxiety, increasing perception, and coping.
So what is going on here? You have to look carefully at research on “EQ” and problem solving to see if IQ was accounted for, or even determined. There is a likelihood that emotional intelligence is similar to creativity in that it tends to increase with intelligence until it reaches a cap. People with IQ’s above 120 (considered the top end of the average range) do not show greater creativity than people with IQ’s at that point. In fact people with IQ’s at the high end have a correlating decrease in creativity. Emotional intelligence also seems to improve with cognitive intelligence up to a point and then apparently decreases. This seems to be particularly noticeable with people that have higher mathematical/logical skills. However, research has yet to determine what that cap point is.
So no matter how much you want to increase cognitive ability, creativity and emotional intelligence to be above average in all areas, you will have to settle for something less. As much as we would like ignore the fact, human intelligence is very finite.